Charles J. Colbourn Editor ADTHM, Lethbridge, Alberta, Canada, July

Springer Proceedings in Mathematics & Statistics

Charles J. Colbourn Editor Algebraic Design Theory and Hadamard Matrices ADTHM, Lethbridge, Alberta, Canada, July 2014 Springer Proceedings in Mathematics & Statistics

Volume 133

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This book series features volumes composed of select contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientiﬁc quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the ﬁeld. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. Charles J. Colbourn Editor

Algebraic Design Theory and Hadamard Matrices ADTHM, Lethbridge, Alberta, Canada, July 2014

123 Editor Charles J. Colbourn School of Computing, Informatics, and Decision Systems Engineering Arizona State University Tempe, AZ, USA

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-319-17728-1 ISBN 978-3-319-17729-8 (eBook) DOI 10.1007/978-3-319-17729-8

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Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com) Preface

A Workshop on Algebraic Design Theory and Hadamard Matrices was held at the University of Lethbridge from 8 July to 11 July, 2014. From 11 July to 13 July, 2014, the follow-on workshop Algebraic design theory with Hadamard matrices: applications, current trends and future directions (14w2199) was held at the Banff International Research Station. Current research and future directions in algebraic design theory and its connections with algebra, computation, communication, cryptography, finite geometry, codes, and physics were examined. At the same time, the workshops celebrated the 70th birthday of Professor Hadi Kharaghani at the University of Lethbridge. Hadi’s contributions to the field have been significant. In 1985, he published a very useful method for constructing some orthogonal matrices, subsequently named Kharaghani matrices. In 1991, he gave the first infinite sequence of Hadamard matrices with maximum excess. In 2000, he constructed one of the largest classes of designs using Kharaghani matrices, introduced twin designs for the first time and developed a new method to generate arrays for orthogonal designs. Together with Behruz Tayfeh-Rezaie, he discovered a Hadamard matrix of order 428 in 2004; the order had been for a long while the smallest order for which no Hadamard matrix was known. Hadi is an author of more than 85 papers published in refereed journals. He is a Foundation Fellow of the Institute of Combinatorics and its Applications and has been named to the editorial board of the Journal of Combinatorial Designs. Hadi has also organized workshops, conferences, and seminars that have promoted research in combinatorics and increased the University of Lethbridge’s profile and reputation. He has won the top research award and the top teaching award at the University. Lastly at an age when many consider retirement, he has taken on the ultimate challenge—administration—by serving as Chair of the largest department at the university! His 6 year sentence ends in a few years. This volume contains selected papers from these two workshops. Many people are to be thanked for their roles in running the workshops and ensuring that the proceedings came to fruition. Thanks to the workshop organizers in Lethbridge (Amir Akbary and Wolf Holzmann, University of Lethbridge; K.T. Arasu and Yuqing Chen, Wright State University; Charlie Colbourn, Arizona State University; Robert

v vi Preface

Craigen, University of Manitoba; and Vladimir Tonchev, Michigan Technological University), and in Banff (Robert Craigen, University of Manitoba; Dane Flannery, National University of Ireland; and Hadi Kharaghani, University of Lethbridge). Thanks also to all of the speakers and participants for making the meetings scientiﬁc successes. Special thanks to Rob Craigen for coordinating the problem section in this proceedings. Finally, thanks to the numerous anonymous reviewers who improved the quality of the papers that you see here. And thanks to Hadi Kharaghani for his many contributions to algebraic design theory and Hadamard matrices. Happy 70th birthday!

Tempe, AZ, USA Charles J. Colbourn January 2015 Contents

On (-1,1)-Matrices of Skew Type with the Maximal Determinant and Tournaments ...... 1 José Andrés Armario On Good Matrices and Skew Hadamard Matrices...... 13 Gene Awyzio and Jennifer Seberry Suitable Permutations, Binary Covering Arrays, and Paley Matrices ..... 29 Charles J. Colbourn Divisible Design Digraphs ...... 43 Dean Crnkovic´ and Hadi Kharaghani New Symmetric (61,16,4) Designs Obtained from Codes ...... 61 Dean Crnkovic,´ Sanja Rukavina, and Vladimir D. Tonchev D-Optimal Matrices of Orders 118, 138, 150, 154 and 174 ...... 71 Dragomir Ž. -Dokovic´ and Ilias S. Kotsireas Periodic Golay Pairs of Length 72...... 83 Dragomir Ž. -Dokovic´ and Ilias S. Kotsireas Classifying Cocyclic Butson Hadamard Matrices ...... 93 Ronan Egan, Dane Flannery, and Padraig Ó Catháin Signed Group Orthogonal Designs and Their Applications ...... 107 Ebrahim Ghaderpour On Symmetric Designs and Binary 3-Frameproof Codes ...... 125 Chuan Guo, Douglas R. Stinson, and Tran van Trung An Algorithm for Constructing Hjelmslev Planes...... 137 Joanne L. Hall and Asha Rao Mutually Unbiased Biangular Vectors and Association Schemes...... 149 W.H. Holzmann, H. Kharaghani, and S. Suda

vii viii Contents

A Simple Construction of Complex Equiangular Lines ...... 159 Jonathan Jedwab and Amy Wiebe Inner Product Vectors for Skew-Hadamard Matrices ...... 171 Ilias S. Kotsireas, Jennifer Seberry, and Yustina S. Suharini Twin Bent Functions and Clifford Algebras ...... 189 Paul C. Leopardi A Walsh–Fourier Approach to the Circulant Hadamard Conjecture ...... 201 Máté Matolcsi A Note on Order and Eigenvalue Multiplicity of Strongly Regular Graphs ...... 209 A. Mohammadian and B. Tayfeh-Rezaie Trades in Complex Hadamard Matrices ...... 213 Padraig Ó Catháin and Ian M. Wanless The Hunt for Weighing Matrices of Small Orders ...... 223 Ferenc Szöllosi˝ Menon–Hadamard Difference Sets Obtained from a Local Field by Natural Projections ...... 235 Mieko Yamada BIRS Workshop 14w2199 July 11–13, 2014 Problem Solving Session ..... 251 R. Craigen (Problems Editor) Contributors

José Andrés Armario Department of Applied Mathematics I, University of Sevilla, Sevilla, Spain Gene Awyzio Faculty of Engineering and Information Sciences, University of Wollongong, Wollongong, NSW, Australia Padraig Ó Catháin School of Mathematical Sciences, Monash University, Melbourne, VIC, Australia Charles J. Colbourn School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, AZ, USA Robert Craigen Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada Dean Crnkovic´ Department of Mathematics, University of Rijeka, Rijeka, Croatia Dragomir Ž. Ðokovic´ Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada Ronan Egan Statistics and Applied Mathematics, School of Mathematics, National University of Ireland, Galway, Ireland Dane Flannery Statistics and Applied Mathematics, School of Mathematics, National University of Ireland, Galway, Ireland Ebrahim Ghaderpour Department of Earth and Space Science and Engineering, York University, Toronto, ON, Canada Chuan Guo David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada Joanne L. Hall School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, Australia Wolfgang Holzmann Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, Canada

ix x Contributors

Jonathan Jedwab Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada H. Kharaghani Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, Canada Ilias S. Kotsireas Department of Physics & Computer Science, Wilfred Laurier University, Waterloo, ON, Canada Paul C. Leopardi Mathematical Sciences Institute, The Australian National University, Canberra, ACT, Australia Máté Matolcsi Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary A. Mohammadian School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran Asha Rao School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, VIC, Australia Sanja Rukavina Department of Mathematics, University of Rijeka, Rijeka, Croatia Jennifer Seberry SCSSE, Centre for Computer and Information Security Research, University of Wollongong, Wollongong, NSW, Australia Douglas R. Stinson David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada S. Suda Department of Mathematics Education, Aichi University of Education, Kariya, Aichi, Japan Yustina S. Suharini Department of Informatics Engineering, Institute of Technology Indonesia, Tangerang, Banten, Indonesia Ferenc Szöllosi˝ Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan B. Tayfeh-Rezaie School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran Vladimir D. Tonchev Department of Mathematical Sciences, Michigan Techno- logical University, Houghton, MI, USA Tran van Trung Institute for Experimental Mathematics, University of Duisburg- Essen, Essen, Germany Ian M. Wanless School of Mathematical Sciences, Monash University, Melbourne, VIC, Australia Contributors xi

Amy Wiebe Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada Mieko Yamada School of Arts and Sciences, Tokyo Woman’s Christian University, Tokyo, Japan On .1; 1/-Matrices of Skew Type with the Maximal Determinant and Tournaments

José Andrés Armario

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract Skew Hadamard matrices of order n give the solution to the question of ﬁnding the largest possible n by n determinant with entries ˙1 of skew type when n Á 0.mod 4/. Characterizations of skew Hadamard matrices in terms of tournaments are well-known. For n Á 2.mod 4/, we give a characterization of .1; 1/-matrices of skew type of order n where their determinants reach Ehlich–Wojtas’ bound in terms of tournaments.

Keywords Tournaments • Maximal determinants • Skew .1;1/-matrices

1 Introduction

Let g.n/ denote the maximum determinant of all n n matrices with elements ˙1. Here and throughout this paper, for convenience, when we say determinant we mean the absolute value of the determinant. The question of ﬁnding g.n/ for any integer n is an old one which remains unanswered in general. We ignore here the trivial cases n D 1; 2. In 1893 Hadamard gave the upper bound nn=2 for g.n/. This bound can be attained only if n is a multiple of 4. A matrix that attains it is called a Hadamard matrix, and it is an outstanding conjecture that one exists for any multiple of 4.At the time of writing, the smallest order for which the existence of a Hadamard matrix is in question is 668. If n is not a multiple of 4, g.n/ is not known in general, but tighter bounds exist. For n Á 2.mod 4/, Ehlich [3] and independently Wojtas [8] proved that

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. J.A. Armario () Department of Applied Mathematics I, University of Sevilla, Sevilla, Spain e-mail: [email protected]

© Springer International Publishing Switzerland 2015 1 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_1 2 J.A. Armario

1 1 g.n/ Ä .2n 2/.n 2/ 2 n (Ehlich–Wojtas’ bound): (1)

This bound can be attained only if 2n 2 is the sum of two squares. When an n n determinant is found that attains the upper bound, it is immediate that the maximal determinant for that order is just the bound itself. From now on, we will call the matrices attaining Ehlich–Wojtas’ bound, E–W matrices. It has been conjectured that E–W matrices of order n exist when 2n 2 D ˛2 C ˇ2 for some positive integers ˛ and ˇ. The interested reader is addressed to [4] and the website [6]for further information on what is known about maximal determinants. A .1; 1/-matrix M of order n is said to be of skew type if M C MT D 2I,where I denotes the identity matrix and MT the transpose of M. A Hadamard matrix H of skew type is called a skew Hadamard matrix. Whenever a skew .1; 1/-matrix is mentioned in this paper, we mean a .1; 1/-matrix of skew type. It was conjectured that skew Hadamard matrices exist for any order multiple of 4. However, it was proved [1] that skew E–W matrices may only exist when 2n 3 D ˛2 for some integer ˛ (i.e., ˇ D 1), a condition which is believed to be sufﬁcient. In [1] examples of skew E–W matrices for small orders have been provided. A tournament T D .V; E/ of order n is a directed graph where the vertex set V consists of n elements and the edge set E V V such that each pair of vertices x and y is joined by exactly one of the edges .x; y/ or .y; x/.Theadjacency matrix A of a directed graph, G D .V; E/, is indexed by the vertex set V, and its entries are deﬁned as follows: 1 if .x; y/ 2 E, ŒA ; D x y 0 otherwise.

Thus, a directed graph T is a tournament if and only if its adjacency matrix satisﬁes

A C AT D J I; (2) where J is the all-ones matrix of order n. Throughout this paper we use A for the adjacency matrix of the tournament T. The vector s D A1 is called the score vector of the tournament, where 1 is the all-ones column vector. Clearly, Â Ã T n ; 1 s D 2 (3) and the i-th entry of s, denote by si,istheoutdegree of vertex i-th in the tournament T. A tournament T of order n is regular if all entries of the score vector .n 1/ n T are equal to 2 , which implies that must be odd. A tournament of even n .n2/ order n is almost regular if the entries of the column vector A1 are 2 and 2 , each n appearing 2 times. The tournament T is doubly regular of degree t provided any two Maximal Determinants and Tournaments 3 vertices of T jointly dominate precisely t vertices. It is easy to see that if T is doubly regular, then T is regular with degree 2t C 1. Thus, T is doubly regular of degree t T if and only if A satisfies AA D tJn C .t C 1/In where n D 4t C 3. Consequently, T is doubly regular with parameters .4t C 3; 2t C 1; t/. C Let ex;y denote the edge of T defined by the vertices x and y. d .ex;y/ denotes the number of vertices dominated by both x and y and d .ex;y/ denotes the number of C vertices dominating both vertices x and y.Thatis,d .ex;y/ [respectively d .ex;y/]is the number of C1’s (respectively, 0’s) that the rows of A indexed by x and y contain in the same column. For a .1; 1/-matrix of skew type M of order n, we normalize M so that the first row of M consists of the all-ones vector with keeping to be of skew type. We can 1 construct a .0; 1/-matrix A of order n 1 as the submatrix of 2 .J M/ obtained by deleting the first row and column. It is easy to see that A satisfies (2), thus A is the adjacency matrix of a tournament of order n 1. Assuming that M is a skew Hadamard matrix in the above construction, it has been shown that the existence of the following are equivalent: 1. Skew Hadamard matrices of order n. 2. Doubly regular tournaments of order n 1 [7]. 3. Tournaments of order n 1 with eigenvalues n2 (with algebraic multiplicity p ! 2 1 1 ˙ n n2 one) and 2 (each with algebraic multiplicity 2 )[2].

Recently, another characterization of a skew Hadamard matrix of order n in terms of the spectrum of the Seidel matrix of a tournament of order n 2 was given in [5]. In this paper, a characterization of skew E–W matrices in terms of tournament is given. This characterization can be considered as the analogous one to the characterization of skew Hadamard matrices given in [7].

Notation. Throughout this paper we use sx for the entry of the score vector (sum h row) corresponding to the row of A indexed by vertex x and V Dfy 2 V W sy D hg. The notation e 2 W Z means an edge of T D .V; E/ deﬁned by one vertex in W and the other one in Z where W and Z are subsets of V.Weuset for a positive integer.

2 The Main Result

As introduced in Sect. 1, Ehlich–Wojtas’ bound provides an upper bound of the maximum determinant of all nn matrices with elements ˙1 when n Á 2.mod 4/. Moreover, equality in (1) holds if and only if there exists a .1; 1/-matrix B of order n, such that Â Ã L 0 BBT D BT B D ; (4) 0 L 4 J.A. Armario

. 2/ 2 n . 1; 1/ with L D n I C J a square matrix of order 2 . In addition, any -matrix M attaining Ehlich–Wojtas’ bound is equivalent to a matrix B satisfying (4)(see[3]). Two matrices M and N are said to be Hadamard equivalent or equivalent if one can be obtained from the other by a sequence of the operations: • interchange any pairs of rows and/or columns; • multiply any rows and/or columns through by 1. In other words, we say that M and N are equivalent if there exist monomial matrices P and Q such that PMQ D N. Let us point out that if M is equivalent to B then B D PMQ where P and Q are monomial matrices. Thus,

BBT D PMQQT MT PT D PMMT PT D .PMPT /.PMPT /T :

Hence, if B satisfies (4), PMPT as well. Remark 2.1. Without loss of generality, we can always assume that if M reaches Ehlich–Wojtas’s bound then there exists a monomial matrix P such that PMPT satisfies (4). Condition (4) implies some combinatorial properties, regarding the number of positive entries of the rows (resp. columns) of B. The rows of any .1; 1/-matrix of size n can be classified as of even or odd type, depending on the parity of the number of C1’s that they contain. It is apparent that the inner product of two rows of the same type is congruent to 2 modulo 4, while the inner product of two rows of opposite type is congruent to 0 modulo 4. In these circumstances, the block structure of the matrix in (4) implies that the rows from 1 to 2t C 1 of B share a common type, whereas the rows from 2t C 2 to 4t C 2 share the opposite type. The same argument translates to the columns of B. This is a main difference with usual Hadamard matrices of order a multiple of 4, in which rows of different type cannot occur. Notice that this balanced structure of even and odd type rows does not need to be attained anymore when 2n 2 is not the sum of two squares. If B is skew, then 2n 2 is the sum of two squares where the first integer is 1 (see [1]). Moreover, assuming that B is normalized, the rows of odd type can be classified into two kinds depending on their row sum and there are exactly the same number of every kind. Lemma 2.1. Let M be a skew .1; 1/-matrix of order n D 4t C 2 and A be its corresponding adjacency matrix of the tournament of order n 1. If M is a skew E–W matrix, then the entries of the score vector A1 are 2t, 2t C 1 and 2t 1,each appearing 2t C 1, t and t times, respectively.

Proof. Let M D Œmi;j be a skew E–W matrix of order n D 4t C 2. We can always assume that the ﬁrst row of M consists entirely of C1 since multiplying row r and column r of M by 1 the value of the determinant does not change and neither does the skew character. This implies mi;1 D1; 2 Ä i Ä n.AsM is skew, mi;i DC1; 1 Ä i Ä n. Form a new matrix A D Œai;j, 1 Ä i; j Ä n 1, by putting Maximal Determinants and Tournaments 5 C1; miC1;jC1 D1 ai;j D 0; miC1;jC1 DC1:

As it was mentioned in Sect. 1, A is the adjacency matrix of a tournament of order n 1 D 4t C 1.SinceM is equivalent to a matrix B satisfying (4), thus for j ¤ 1, Xn Xn ˙2; 2 Ä j Ä 2t C 1; m1; m ; D m ; D i j i j i 0; 2t C 2 Ä j Ä 4t C 2I iD1 iD1 so that each row of M from the 2tC2-th until the 4tC2-th contains .2tC1/ C1’s and .2tC1/ 1’s. Thus, the last 2tC1 rows of A contains exactly 2t C1’s, so that the last 2t C 1 entries of the score vector are equal to 2t (i.e., si D 2t;2t C 1 Ä i Ä 4t 1). For the rows of M from the 2-nd until the 2t C 1-th, we have two cases: Xn 1. If mj;i DC2, then every type of these rows of M contains .2t C 2/ C 1’s and iD1 .2t/ 1’s. Thus, the corresponding rows of A contain .2t 1/ C 1’s. Xn 2. If mj;i D2, then every type of these rows of M contain .2t/ C 1’s and iD1 .2t C 2/ 1’s. Thus, the corresponding rows of A contain .2t C 1/ C 1’s. So that the ﬁrst 2t entries of the score vector are equal to either 2t 1 or 2t (i.e., si D 2t C 1 or si D 2t 1; 1 Ä i Ä 2t). 2tC1 2t1 Finally, let jV jDjfiW si D 2t C 1gj and jV jDjfiW si D 2t 1gj.Wehave ( jV2tC1jCjV2t1jD2t; jV2tC1j.2t C 1/ CjV2t1j.2t 1/ D 4t2:

The second equation of the system above follows from (3). Hence, jV2tC1jD jV2t1jDt, and this concludes with the desired result. ut Lemma 2.2. Assume the same hypothesis and notation as in Lemma 2.1. Thus, the adjacency matrix A satisﬁes (after permutations of row and columns) Ä Ä XY XQ YQ AAT D and AT A D (5) ZW ZWQ where Ä Ä X1 X2 X4 X2 • X D and XQ D with X3 X4 X3 X1

X1 D tIt C .t 1/Jt; X4 D tIt C .t C 1/Jt; X2 D X3 D .t 1/Jt: 6 J.A. Armario Ä Ä .t 1/J ;2 1 tJ;2 1 • Y D t tC and YQ D t tC : tJt;2tC1 .t 1/Jt;2tC1

• Z D YT and ZQ D YQ T :

Ä W1 W2 • W D with W3 W4

T W1 D tI C tJ ; W2 D W3 D .t 1/J;; W4 D tI C tJ ;

where C D 2t C 1 and ; 0.Jr;s denotes the all-ones matrix with r rows and s columns, and Jr denotes Jr;r. We follow the same notation for the identity matrix. Proof. Considering the rows .k 1/-th and .l 1/-th of A, with k < l.Let

˛ Djfi W mk;i D ml;i DC1gj; ˇ Djfi W mk;i D ml;i D1gj; Djfi W mk;i Dml;i DC1gj; ı Djfi W mk;i Dml;i D1gj:

By construction of A, ˇ 1 (respectively, ˛) is equal to the number of C1’s (respectively, 0’s) that rows .k1/ and .l1/ of A contain in the same column. That is, the vertices .k 1/ and .l 1/ of T jointly dominate precisely ˇ 1 vertices and, they are jointly dominated precisely by ˛1 vertices (since either mk;l D ml;l DC1 or mk;k D ml;k DC1). Therefore,

T T ŒAA k1;l1 D ˇ 1 and ŒA Ak1;l1 D ˛ 1: (6)

Now considering the set of rows of M but the ﬁrst, we can classify them attending to their row sum in three types. For 2 Ä r Ä 4t C 2

4XtC2 R1 DfrW mr;i D2g; iD1 4XtC2 R2 DfrW mr;i D 0g; iD1 4XtC2 R3 DfrW mr;i DC2g: iD1

In Lemma 2.1 it was proven

1. r 2 R2 , 2t C 2 Ä r Ä 4t C 2. Maximal Determinants and Tournaments 7

2. jR1jDjR3jDt: Attending to this classiﬁcation, there are six possible cases of choosing pairs of rows.

1. Let k 2 R1 and l 2 R2. Counting the 1 and C1 in each row. We have

˛ C ı D 2t C 1 D ˇ C ˇ C ı D 2t C 2 ˛ C D 2t

so ˛ C ı D ˇ C D ˇ C ı 1 D ˛ C C 1,orı D C 1 and ˇ D ˛ C 1. Recall that negating certain set of rows and the sameP set of columns of M, 4tC2 0 we obtain a new matrix B which satisﬁes (4). Hence, iD1 mk;iml;i D ,so ˛ C ˇ D C ı. Thus, ˇ D t C 1 and ˛ D t. 2. Let k 2 R3 and l 2 R2. By a similar argument, ˇ D t and ˛ D t C 1. 3. Let k; l 2 R2. Counting the number of 1 and C1 in each row,

ˇ C ı D ˛ C D 2t C 1 D ˇ C D ˛ C ı; P ı ˛ ˇ 2 so DP and D .Also, i mk;iml;i D˙ . 2 If i mk;iml;i DC ,then

˛ C ˇ D 2t C 2 C ı D 2t:

Thus ˇ D t C 1PD ˛. 2 ˇ ˛ Similarly, if i mk;iml;i D ,then D t D . 4. Let k 2 R1 and l 2 R3. Counting the number of 1 and C1 in each row, we get ˛ D ˇ. Since negating certain set of rows andP the same set of columns of M,we 2 ˛ ˇ 2 obtain a new matrix B which satisﬁes (4), then i mk;iml;i D .So, C D t. Hence, ˇ D t D ˛. P ; 2 5. Let k l 2 R1. Taking into account that i mk;iml;i D and using an analogous argument. We get ˇ D t C 2 and ˛ D t.P ; 2 6. Let k l 2 R3. Taking into account that i mk;iml;i D and using an analogous argument. We get ˇ D t and ˛ D t C 2.

Consequently, taking into account the size of the sets R1; R2,andR3, the identities (6) and the values of ˛ and ˇ, it follows the desired result. ut Deﬁnition 2.1. Let T D .V; E/ be a tournament of order 4t C 1,itsaidtobean E–W tournament if • its adjacency matrix A satisﬁes (after permutations of row and columns) the identities in (5). Or equivalently, 8 J.A. Armario

• The entries of its score vector si are 2t, 2t C 1 and 2t 1, each appearing 2t C 1, t and t times, respectively. •Lete 2 E. 8 2t 2tC1 ˆ .t; t 1/; e 2 V V ; ˆ 2t1 2t ˆ .t 1; t/; e 2 V V ; ˆ 2 1 2 1 ˆ .t 1; t 1/; e 2 V t V tC ; ˆ ˆ .t C 1; t 1/; e 2 V2tC1 V2tC1; <ˆ .t 1; t C 1/; e 2 V2t1 V2t1; .dC; d/.e/ D ˆ .t; t/; e 2fxgV ; ; ˆ x t ˆ .t 1; t 1/; e 2fxgV ; 1; ˆ x t ˆ . ; /; ; ˆ t t e 2 Vx;t Vx;t ˆ . ; /; ; :ˆ t t e 2 Vx;t1 Vx;t1 .t 1; t 1/; e 2 Vx;t1 Vx;tI

2t 2t C for some x 2 V ; let Vx;d Dfy 2 V W d .ex;y/ D dg such that Vx;t [ Vx;t1 D 2t V nfxg. Moreover, either Vx;t or Vx;t1 could be empty. Remark 2.2. The equivalence above follows from the fact that the vertices .k 1/ T and .l 1/ of T jointly dominate precisely ŒAA k1;l1 vertices and, they are jointly T dominated precisely by ŒA Ak1;l1 vertices. The following result follows immediately from Lemmas 2.1 and 2.2. Proposition 2.1. Let M be a skew .1; 1/-matrix of order n D 4t C 2 and A be its corresponding adjacency matrix of the tournament of order n 1. If M is a skew E–W matrix, then the tournament with adjacency matrix A is an E–W tournament. In the next result, we will show that the converse statement holds.

Proposition 2.2. Let T be a tournament of order 4t C 1 and A D Œai;j be its adjacency matrix. If T is an E–W tournament, then M D Œmi;j with 8 < C1; i D 1; 1; >1; 1 mi;j D : i and j D 1 2ai1;j1;2Ä i; j Ä 4t C 2: is a skew E–W matrix of order 4t C 2. Proof. We can always assume that the vertices of V are ordered in such a way so, 8 < 2t 1; 1 Ä i Ä t; 2 1; 1 2 ; si D : t C t C Ä i Ä t 2t;2t C 1 Ä i Ä 4t C 1: Maximal Determinants and Tournaments 9

We have to show that M is equivalent to B such that B satisﬁes (4). To this end, consider sums of the form

4XtC2 mk;iml;i;1Ä k; l Ä 4t C 2: iD1

1. k D l,

4XtC2 4XtC2 4XtC2 2 1 4 2: mk;iml;i D mk;i D D t C iD1 iD1 iD1

2. For k D 1 ¤ l, 8 4XtC2 4XtC2 < C2; 2 Ä l Ä t C 1; 4 2 2; 2 2 1: m1;iml;i D ml;i D t sl1 D : t C Ä l Ä t C iD1 iD1 0; 2t C 2 Ä l Ä 4t C 2:

3. 2 Ä k Ä 2t C 1 and 2t C 2 Ä l Ä 4t C 2. C. / 1 . / 1 Recall that d exk;xl Djfi W ak;i D al;i DCgj and d exk;xl C D 0 . C; /. / . ; 1/ jfi W ak;i D al;i D gj. Since either d d exk1;xl1 D t t or . C; /. / . 1; / 1 2 1 d d exk1;xl1 D t t ,sojfi W mk;iml;i D gj D t C . Thus,

4XtC2 mk;iml;i D .2t C 1/ .2t C 1/ D 0: iD1

2 1 2 2 1 . C; /. / . 1; 1/ 4. Ä k Ä t C and t C Ä l Ä t C .Since d d exk1;xl1 D t t , so jfi W mk;iml;i D 1gj D 2t. Thus,

4XtC2 mk;iml;i D 2t .2t C 2/ D2: iD1

5. 2 Ä k < l Ä t C 1 or t C 2 Ä k < l Ä 2t C 1. In similar manner, we have

4XtC2 mk;iml;i D .2t C 2/ 2t D 2: iD1 2 2 < 4 2 6. t C Ä k l Ä t C . Let us take the sets Vx2tC1;t and Vx2tC1;t1. Consider the cases:

(a) For k D 2t C 2. P 4tC2 2: If xl1 2 Vx2tC1;t. Thus, iD1 mk;iml;i D 10 J.A. Armario P 4tC2 2: If xl1 2 Vx2tC1;t1. Thus, iD1 mk;iml;i D (b) For 2t C 2

(i) If exk1;xl1 2 Vx2tC1;t Vx2tC1;t Vx2tC1;t1 Vx2tC1;t1 . Thus,

4XtC2 mk;iml;i D 2: iD1

(ii) If exk1;xl1 2 Vx2tC1;t1 Vx2tC1;t. Thus,

4XtC2 mk;iml;i D2: iD1

Consequently, multiplying by 1 the rows and columns of M from the .t C 2/-th until the .2t C 1/-th and the corresponding rows and columns of M indexed with the elements of Vx2tC1;t1, then this new matrix denoted by B satisﬁes (4). By construction it is easy to see that MCMT D 2I. This concludes the proof. ut Finally, we state the main result of this paper which is an immediate consequence of Propositions 2.1 and 2.2. Theorem 2.1. The existence of the following are equivalent: 1. Skew E–W matrices of order 4t C 2. 2. E–W tournaments of order 4t C 1.

3 Conclusions and Further Work

In this paper we have proved that the existence of a skew E–W matrix of order 4t C 2 is equivalent to the existence of an E–W tournament of order 4t C 1.This kind of tournament has been deﬁned attending to their vertices and edges degrees. Examples of skew E–W matrices for small orders have been provided in [1]. A characterization of E–W matrices of order 4t C 2 in terms of spectral data for tournaments of order 4t C 1, as analogous to the result for skew Hadamard matrices given in [2], is a challenging problem. After looking over small orders, we conjecture that the existence of the following are equivalent: 1. Skew E–W matrices of order n D 4t C 2. 2. Tournaments of order n 1 with eigenvalues: n 4 (a) The roots of the polynomial P.x/ Dt.n 3/ x2 C x3. p 2 1 ˙ 3 n n 4 (b) 2 each with algebraic multiplicity 2 . Maximal Determinants and Tournaments 11

Acknowledgements The author would like to thank the anonymous referee for his valuable comments. The author would also like to thank Kristeen Cheng for her reading of this manuscript. This work has been partially supported by the research project FQM-016 from JJAA (Spain).

References

1. Armario, J.A., Frau, M.D.: An upper bound on the maximal determinant of skew matrices of the conference type (2014, preprint) 2. de Caen, D., Gregory, D.A., Kirkland, S.J., Pullman, N.J.: Algebraic multiplicity of the eigenvalues of a tournament matrix. Linear Algebra Appl. 169, 179–193 (1992) 3. Ehlich, H.: Determiantenabschätzungen für binäre Matrizen. Math. Z. 83, 123–132 (1964) 4. Kharaghani, H., Orrick, W.: D-optimal matrices. In: Colbourn, C., Dinitz, J. (eds.) The CRC Handbook of Combinatorial Designs, pp. 296–298, 2nd edn. Taylor and Francis, Boca Raton (2006) 5. Nozaki, H., Suda, S.: A characterization of skew Hadamard matrices and doubly regular tournaments. Linear Algebra Appl. 437, 1050–1056 (2012) 6. Orrick, W., Solomon, B.: The hadamard maximal determinant problem (2005). http://www. indiana.edu/~maxdet/. Cited 3 Sept 2014 7. Reid, K.B., Brown, E.: Doubly regular tournaments are equivalent to skew Hadamard matrices. J. Comb. Theory Ser. A 12, 332–338 (1972) 8. Wojtas, W.: On Hadamard’s inequallity for the determinants of order non-divisible by 4. Colloq. Math. 12, 73–83 (1964) On Good Matrices and Skew Hadamard Matrices

Gene Awyzio and Jennifer Seberry

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract In her Ph.D. thesis (Seberry) Wallis described a method using a variation of the Williamson array to find suitable matrices, which we will call good matrices, to construct skew Hadamard matrices. Good matrices were designed to plug into the Seberry–Williamson array to give skew-Hadamard matrices. We investigate the properties of good matrices in an effort to find a new, efficient, method to compute these matrices. We give the parameters of the supplementary difference sets (SDS) which give good matrices for use in the Seberry–Williamson array.

Keywords Hadamard matrices • Seberry-Williamson array • skew-Hadamard matrices • Good matrices • Supplementary difference sets • 05B20

1 Introduction

Many constructions for ˙1 matrices and similar matrices such as Hadamard matrices, weighing matrices, conference matrices and D-optimal designs use skew Hadamard matrices in their construction. For more details, see Seberry and Yamada [21]. A Hadamard matrix is a square matrix with elements of ˙1 and mutually orthogonal rows. Thus a 4w 4w Hadamard matrix must have 2w.4w 1/ entries

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. G. Awyzio Faculty of Engineering and Information Sciences, University of Wollongong, Wollongong, NSW 2522, Australia e-mail: [email protected] J. Seberry () SCSSE, Centre for Computer and Information Security Research, Faculty of Engineering and Information Sciences, University of Wollongong, Wollongong, NSW 2522, Australia e-mail: [email protected]

© Springer International Publishing Switzerland 2015 13 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_2 14 G. Awyzio and J. Seberry of 1 and 2w.4w C 1/ entries of C1 for a normalized Hadamard matrix, that is one where the first row and first column are all C1. For any Hadamard matrix H of size > > 4w 4wHH D 4wI4w D H H. In all our examples minus (“”) is used to denote minus one (“1”). Hadamard matrices of order 4w, which first arose in [22], all orders 4w Á 0 .mod 4/. A weighing matrix, W D W.4w; k/,oforder4w and weight k has elements > 0; ˙1 and satisfies WW D kI4w. These are conjectured to exist for all k D 1 4w for each order 4w. If a Hadamard matrix M can be written in the form M D I C S where S> DS,thenM is said to be a skew-Hadamard matrix. Skew-Hadamard matrices are also conjectured to exist for all orders 4w Á 0.mod 4/.Thefirst unresolved case for Hadamard matrices is currently 4 167. Example 1.1 (Hadamard Matrices). 2 3 2 3 1111 1111 Ä 11 6 11 7 6 11 7 6 7 6 7 : H2 D H4symmetric D 4 5 H4skew-type D 4 5 1 1 1 11 1 1 1 1

1.1 Circulant and Type 1 Matrix Basics

Because it is so important for the rest of our work we now spend a little effort to establish why the properties required for Williamson matrices are so important. We deﬁne the shift matrix, T of order n by 2 3 010 6 7 6 000 7 T D 6 : : 7 : 4 : : 5 100

So any circulant matrix, of order n and ﬁrst row x1; x2; ; xn,thatis, 2 3 x1 x2 x3 xn 6 7 6 xn x1 x2 xn1 7 6 7 6 xn1 xn x1 xn2 7 (1) 6 : : 7 4 : : 5 x2 x3 x4 x1 can be written as the polynomial

n 2 n1 x1T C x2T C x3T xnT : On Good Matrices and Skew Hadamard Matrices 15

We now note that polynomials commute, so any two circulant matrices of the same order n commute. We deﬁne the back-diagonal matrix, R of order n by 2 3 001 6 7 6 000 7 R D 6 : : 7 : 4 : : 5 100

Since TmR is the polynomial for any integer m 0, we have that, similarly, any back-circulant matrix, of order n and ﬁrst row x1; x2; ; xn,thatis, 2 3 x1 x2 x3 xn 6 7 6 x2 x3 x4 x1 7 6 7 6 x3 x4 x5 x2 7 6 : : 7 4 : : 5

xn x1 x2 xn1 can be written as the polynomial

n 2 n1 x1T R C x2TR C x3T R xnT R:

Mathematically we have that A circulant matrix C D .cij/ of order n is a matrix which satisfies the condition that cij D c1;jiC1,wherejiC1 is reduced modulo n [25]. A back-circulant matrix B D .bij/ order n is a matrix with property that bij D b1;iCj1,wherei C j 1 is reduced modulo n [25]. The transpose of a back-circulant matrix is the same as itself, so it is also a symmetric matrix. In this paper we will not need to study back-circulant matrices further but define them here for completeness only. In all our definitions of circulant and back-circulant matrices we have assumed that the rows and columns have been indexed by the order, that is for order n,the rows are named after the integers 1; 2; ; n and similarly for the columns. The internal entries are then defined by the first row using a 1:1 and onto mapping f W G ! G. However we could have indexed the rows and columns using the elements of a group G, with elements g1, g2, , gn. Loosely a type one matrix will then be defined so the .ij/ element depends on a 1:1 and onto mapping of f .gj gi/ for type 1 matrices and of f .gj C gi/ for type two matrices. We use additive notation, but that is not necessary. Wallis and Whiteman [29] have shown that circulant and type 1 can be used interchangeably in the enunciations of theorems as can the terms back-circulant and type 2. This can be used to explore similar theorems in more structured groups. 16 G. Awyzio and J. Seberry

2 Historical Background

Hadamard matrices first appeared in the literature in an 1867 paper written by Sylvester [22]. In 1892 Hadamard matrices first appear. They were called matrices on the unit circle as they satisfied Hadamard’s inequality for the determinant of matrices with entries within the unit circle [13]. Later Scarpis [19] found many Hadamard matrices using primes. In 1933 Paley [18] conjectured that Hadamard matrices existed for all positive integer orders divisible by 4. This has become known as the Hadamard conjecture: Conjecture 2.1 (Paley). Hadamard matrices exist for all orders 1, 2, 4w,wherew is a positive integer. Paley’s work [18, 24] left many orders for Hadamard matrices unresolved. Later Williamson [30] gave a method that many researchers hoped would give results for all orders of Hadamard matrices. Many results are given in [1–4]. That the Williamson method would give results for all orders of Hadamard matrices was first disproved by -Dokovic´ in 1993 [7]. Holtzmann et al.[15] showed that we have no Williamson constructions for some smallorders. They give the following table (Table 1):

Table 1 Number of Williamson matrices of order 1–59 Order: 1357911131517192123252729 Number: 1112314446711061 Order: 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Number: 250411210120110

Good matrices ﬁrst appeared in the Ph.D. Thesis of Jennifer (Seberry) Wallis [24]. There the matrices, which were given no name, were given for w D 1; 15; 19. In 1971 she gave good matrices for w D 23 [23]. The array and construction using the Seberry–Williamson array to construct skew-Hadamard matrices was also given there, but not named. Good matrices were ﬁrst used by name in [27]. Hunt [16] gave the matrices for w D 1; ;25. Later Szekeres gave a list for order w D 1; ;31. -Dokovic[´ 7, 8] provided orders w D 33; 35 and 127. Then Georgiou et al. [12] provided 37, 39. -Dokovic[´ 11] says that only one set of supplementary difference sets (SDS), (41;20,20,16,16;31), for 41 remains to be searched. We note that while there are no Williamson matrices of order 35 there are good matrices of order 35. (Seberry) Wallis [24] gave a construction for w D 19 and -Dokovic[´ 8]forw D 33; 35, and 127. The remainder were found by computer search. Other relevant publications are: [5, 6, 9–11, 20, 28]. On Good Matrices and Skew Hadamard Matrices 17

3 Williamson Type Constructions

3.1 Williamson Array

In 1944 [30] Williamson proposed using what has come to be known as the Williamson array. If we can calculate suitable matrices of order w, they can be plugged-in to his array to give Hadamard matrices of order 4w. We use the Williamson-array in the form 2 3 ABCD 6 7 6 BADC 7 WWilliamson-array D 4 5 C DAB DCBA where A; B; C; D are circulant (cyclic) matrices with symmetric ﬁrst rows which satisfy the additive property

> > > > AA C BB C CC C DD D 4wIw: (2)

These matrices are known as Williamson matrices The Williamson-array is formally orthogonal. A Hadamard matrix is made by plugging Williamson matrices into the Wwilliamson-array. Example 3.1 (Williamson Matrix of Order 4 3). 2 3 1 1 1 111 6 7 6 1 1 1 1117 6 7 6 1 1 1 1117 6 7 6 111 111 117 6 7 6 7 6 1 1 1 1111 1 7 6 7 6 11 1 11111 7 W12 D 6 7 Williamson 6 11 1 1 7 6 7 6 1 1 1 1 7 6 7 6 11 1 1 7 6 7 6 7 6 1 111 7 4 1 1 1 1 5 1 11 1

Remark 3.1. An example of the crucial part of proof is: When we look at the terms > of Wwilliamson-arrayWwilliamson-array for, say the (2,3) element we have

BC> AD> C DA> CD> D BC AD C DA CB D 0: 18 G. Awyzio and J. Seberry

Noting matrices A, B, C and D are polynomials, and commute, the (2,3) element will reduce to zero. Similarly the other off diagonal elements are 0. ut

3.2 Seberry–Williamson Array

In her Ph.D. thesis [24] Seberry Wallis gave a Seberry–Williamson array, a modification of the Williamson array which can be used to make skew Hadamard matrices. Written in terms of circulant and back circulant matrices it is 2 3 ABRCRDR 6 7 6 BR A DR CR 7 WSeberry–Williamson-array D 4 5 ; CR DR A BR DR CR BR A where A is a ˙1 skew-type cyclic matrix and B; C; D are circulant matrices with symmetric first rows that satisfy the additive property given in Eq. (2). These are called good matrices [27]. Thus the first rows each have the form

A D .1/.SA/.SA /; B D .1/.SB/.SB /; C D .1/.SC/.SC /; D D .1/.SD/.SD /; where S means the elements of S in reverse order. SX are the elements of SX reversed. (In matrix terms this is RXR.) A skew-Hadamard matrix can be made by plugging good matrices into the WSeberry–Williamson-array. Example 3.2 (Seberry–Williamson Matrix of Order 43). The Seberry–Williamson matrix for ﬁrst rows A D 11, B D 1 , C D 1 , D D 111 is 2 3 11 1 1 111 6 7 6 11 1 1 1117 6 7 6 1 1 1 1 1117 6 7 6 11 11 11111 7 6 7 6 7 6 1 1 111111 1 7 6 7 6 111 1 111 117 WSeberry–Williamson D 6 7 : 6 11 11 1 7 6 7 6 1 1 11 1 7 6 7 6 11 1 1 1 7 6 7 6 7 6 1 11 11 7 4 1 1 1 115 1 111 1 On Good Matrices and Skew Hadamard Matrices 19

3.3 Structure of First Rows of Williamson and Good Matrices

For the Williamson Hadamard matrix each of the four matrices A, B, C and D,of order w, w odd, is symmetric and thus can be written with ﬁrst row

D 1; x1; x2; ; x w1 ; x w1 ; ; x2; x1; „ ƒ‚ 2… „ 2 ƒ‚ … q negative elements q negative elements

1 ; w1 1 where each xi, i D , 2 is ˙ . We will refer to these ﬁrst rows as

Df1; S; S g : where, for example, the ﬁrst row of A,

A Df1; SA; SA g contains

S D x1; x2; ; x w1 A 2 and SA , the reverse of this sequence, is:

S Dx w1 ; ; x2; x1 : A 2

Similarly we obtain SB, SB , SC, SC , SD and SD . The number of such matrices, w1 in each case, 2 2 . We see that the congruence class of SA is given by the two little endian entries of SA. These properties will allow us to impose constraints on the search space (and time) to ﬁnd these matrices.

Lemma 3.1. Suppose that A is a circulant matrix with first row written as A D .1/. /. / 0 0 .1/. /. / SA SA . Then the matrix A with first row A D SA SA ,thatis,it has all the second to wth elements of the first row written in the reverse order, has exactly the same inner products of its rows as A. From Hall [14, lemma 14.2.1] we have a lemma for Williamson matrices: Lemma 3.2. If w is odd, and if the Williamson matrices A; B; C; D, are chosen so the first element of their first rows is +1, then for each i D 2; ; w exactly three of the ith elements of the first rows have the same sign. 20 G. Awyzio and J. Seberry

3.4 Structure of First Rows of Good Matrices

We now consider further the good matrices, A, B, C, D of order m which satisfy the additive property (2). Write, using the shift matrix T,

A D P1 N1 (3) with P1 the sum of the terms with positive coefﬁcient in A and N1 the sum of the terms with negative coefﬁcient in A, whence X X j1 j1 P1 D a1jT ; a1j DC1; N D a1jT ; a1j D1 (4) j j

In the same way write

B D P2 N2 ; C D P3 N3 ; D D P4 N4 : (5)

Since a11 DC1 and A is circulant and skew-type, a1j Da1;mC2j, 2 Ä j Ä m. Hence there are m C 1 p1 2 D (6) positive terms in the first row of A, so the number of terms in the first row of P1 is an odd number if m Á 1.mod 4/ and an even number if m Á 3.mod 4/. Since B is circulant and symmetric we may choose b11 DC1 and b1j D b1;mC2j, 2 Ä j Ä m. The positive terms occur in pairs, so p2 the number of positive terms in the first row of P2, is an odd number. Similarly p3 and p4 are odd numbers. We now write J D I C T C T2 C :::Tm1 D I C T C T2 :::Tm1 R: (7)

Then

Pi C Ni D J ; i D 1; 2; 3; 4 (8) so the additive property (2) becomes

T 2 2 2 AA C B C C C D D 4mIm and by (3)–(5) this becomes

T 2 2 2 .2P1 J/.2P1 J/ C .2P2 J/ C .2P3 J/ C .2P4 J/ D 4mIm ; On Good Matrices and Skew Hadamard Matrices 21 that is

2 2 2 .2P1 J/.2P1 C J C 2I/ C .2P2 J/ C .2P3 J/ C .2P4 J/ D 4mIm ;

2 since A is skew-type. So we have since PiJ D piJ and J D mJ 2 2 2 2 4 P1 C P2 C P3 C P4 C P1 C 4.p1 p2 p3 p4/ J C .2m 2/ J D 4mIm : (9) Now from (6)

4p1 C 2m 2 D 4m so (9) becomes 2 2 2 2 P1 C P2 C P3 C P4 C P1 D.m p2 p3 p4/ J C mIm : (10)

If m is odd, then since P2, P3 and P4 all have an odd number of positive elements in their ﬁrst rows, the coefﬁcients of J are all even. Now notice B, C, D are polynomials in T and R so 0 1 X @ j1A Pi D e1jT R j

2;3;4 T for i D and e1j D b1j, c1j, d1j, respectively, also Pi D Pi so 0 1 ! X X X 2 T @ j1A T k1 n ; Pi D PiPi D e1jT RR e1kT D fnT j k n

2 2 2 2 that is P1, P2, P3,andP4 may all be regarded as polynomials in T. 1;:::; 1 . s/2 t 2 For each t D m , there is a unique s such that T D T .InPj D P 2 Tk , k is a subset of 1;2;:::;m,wehave X 2 k . 2/: Pj Á T mod

Then since the coefﬁcient of J in (10)isalwayseven,wehaveshown

Theorem 3.1. If m is odd, T the shift matrix, and P1,P2,P3,P4 are the terms with positive coefﬁcients of A, B, C, D as deﬁned by (8), respectively, and if

AAT C BBT C CCT C DDT D 4mI 22 G. Awyzio and J. Seberry then writing X X 2 i 2 2 2 i P1 C P1 D fiT and P2 C P3 C P4 D giT i i i¤0 i¤0 we have gi D fi .mod 2/ when i ¤ 0. Hall’s lemma [14] allowed considerable improvements in algorithms to ﬁnd Williamson matrices. Theorem 3.1 allows for improvements in algorithms to ﬁnd good matrices.

3.5 Sums of Squares of First Rows for Arrays

We notice that for the arrays, of Williamson and Seberry–Williamson, which have ˙1 matrices, A, B, C, D of order w plugged into them, these must satisfy the additive property (2).

> > > > AA C BB C CC C DD D 4wIw:

Then if e is the 1 w matrix of all ones and the row sums of A; B; C; D are a; b; c, and d, respectively. Then

eA D ae ; eB D be ; eC D ce ; eD D de ; and

e.AA> C BB> C CC> C DD>/ D a2e C b2e C c2e C d2e D 4we :

For Williamson and good matrices we have

4w D a2 C b2 C c2 C d2 :

Lemma 3.3. For the a, b, c, d, and w just deﬁned for Williamson matrices

a Á b Á c Á d Á w .mod 4/ :

For good matrices b Á c Á d Á w .mod 4/ : ut On Good Matrices and Skew Hadamard Matrices 23

4 Implications for Seberry–Williamson Arrays and Good Matrices

Note if

4w D a2 C b2 C c2 C d2; (11) b, c, d and w are all of the same congruence class modulo 4. a is always 1 within a good matrix (Table 2).

Table 2 Table of good matrix observations Length Order Pattern of four squares b, c, d b C c C d C w jbjCjcjCjdjCw 21 84 12 C 12 C 12 C d2 1, 1, 9 32 32 5 20 12 C 12 C .c/2 C .c/2 1, 3, 3 0 12 13 52 12 C 12 C c2 C c2 1, 5, 5 24 24 9 36 12 C 12 C c2 C d2 1, 3, 5 12 18 7 28 12 C b2 C b2 C b2 3, 3, 3 16 16 19 76 12 C .b2/ C .b2/ C .b2/ 5, 5, 5 4 34 17 68 12 C b2 C b2 C c2 3, 3, 7 4 30 21 84 12 C b2 C c2 C d2 3, 5, 7 16 36 39 156 12 C b2 C c2 C d2 5, 7, 9 32 60

5 Some Observations

Lemma 5.1. Squares of odd numbers are Á 1.mod 4/. Thus b C c C d C w Á 0 .mod 4/. Lemma 5.2. The only possible sum of four squares with 4w D 12 C.˙1/2 Cc2 Cc2 has w Á c Á 1.mod 4// (c can be negative). Proof. We note that c2 is always congruent to Á 1.mod 4/. Consider the case where 4w D 12C12 C2c2 with c Á 1.mod 4/,andc D 4tC1, then

4w D 1C1C2.4tC1/2 D 2C2.16t2C8tC1/ D 2C32t2C16tC2 D 4C32t2C16t :

So w D 1 C 8t2 C 4t and w Á 1.mod 4/. 24 G. Awyzio and J. Seberry

In the case where 4w D 12 C .1/2 C 2c2 with c Á 3.mod 4/,andc D 4t C 3, then

4w D 1 C 1 C 2.4t C 3/2 D 2 C 2.16t2 C 24t C 9/ D 2 C 32t2 C 48t C 18 D 20 C 32t2 C 48t :

So w D 5 C 8t2 C 12t and w Á 1.mod 4/. Thus c cannot be Á 3.mod 4/. ut Lemma 5.3. If 4w D a2 C b2 C c2 C d2 (where a; b; c; d are all in the same congruence class as w .mod 4/), then the number of ones in the ﬁrst rows of ACBC 1Cw bCw cCw dCw C C D (where A,B, C, D are good matrices) are respectively D 2 ; 2 ; 2 ; 2 and so the total number of ones in the ﬁrst rows of the four good matrices is aCbCcCdC4w 2 .

6 Good Matrices and SDS

Good matrices and any other set of four ˙1 matrices which satisfy the additive property of equation (2) can be used to form SDS (see [17]). Example 6.1. In our example of Seberry–Williamson array we could have used the ﬁrst rows 11-,1--,1--,111for the good matrices. These correspond to sets f1; 2g, f1g, f1g, f1; 2; 3g which are in fact SDS as deﬁned below.

Deﬁnition 6.1. A .v; k;/ difference set .d1; ; dk/ is a subset of v such that all the differences di dj; i; j 2f1; ;v 1g occur precisely times. Note that .v 1/ D k.k 1/. Example 6.2. A (13,4,1) difference set is f0; 1; 3; 9g because the differences

0139 0 139 1 12 28 3 10 11 6 9 457 f1; ;12g each occur once.

We use Wallis’ [26] deﬁnition for n fvI k1; k2; knI g supplementary differences sets such that all the differences di dj .mod n/; i; j 2f1; ; kng occur precisely times. Note

Xn .v 1/ D ki.ki 1/: (12) iD1 On Good Matrices and Skew Hadamard Matrices 25

Example 6.3. 4 f9I 5; 5; 3; 7I 11g SDS are

f1; 2; 3; 5g; f1; 3; 4; 7; 8g; f1; 2; 9g; f1; 2; 3; 5; 6; 8; 9g:

1235689 13478 1 124578 1235 1 2367 129 2 8 13467 1 124 3 7 145 1 18 3 78 2356 2 8 13 4 68 34 2 8 7 5 567 134 3 78 2 7 356 1 9 12 6 4568 27 5 567 8 2458 8 23467 1 9 123528

We note that (12) becomes

11.9 1/ D 88 D 20 C 20 C 6 C 42 :

We now give the relationship between the necessary condition for good matrices from (11)

4w D a2.D 12/ C b2 C c2 C d2 and G DfwI k; k2; k3; k4I gsds. Lemma 6.1. Let w be the order of 4 good matrices. Then, with

4w D a2 C b2 C c2 C d2 where a D 1 and w; b; c; d all in the same congruence class modulo 4 the good matrices correspond to a C w b C w c C w d C w a C b C c C d 4 w ; ; ; w I 2 2 2 2 I C 2 (13)

SDS. Proof. We write the positions of 1s in the ﬁrst rows of the good matrices as subsets of w of size k1, k2, k3, k4 then using p for the number of positive elements and q for the number of negative elements in each sds, i D 2; 3; 4 we have

p C q D w (14) and

p2 q2 D b; p3 q3 D c and p4 q4 D d : (15) 26 G. Awyzio and J. Seberry

1 1 Thus, 2p2 D wCb, 2p3 D wCc, 2p4 D wCd as k2 D 2 .wCb/, k3 D 2 .wCc/ and 1 . / 1 1 . / k4 D 2 w C d .Sincea D and the ﬁrst good matrix is skew-type k1 D 2 w C a . Now for sds 4 fwI k1; k2; k3; k4I g we have

Xn .v 1/ D ki.ki 1/ iD1 and Â ÃÂ Ã Â ÃÂ Ã a C w a C w 2 b C w b C w 2 .w 1/ D 2 2 C 2 2 Â ÃÂ Ã Â ÃÂ Ã c C w c C w 2 d C w d C w 2 C 2 2 C 2 2 (16)

1 a2 w2 2aw 2w 2a b2 w2 2bw 2w 2b c2 w2 D 4 C C C C C C C C cbw 2w 2c C d2 C w2 C 2dw 2w 2d

1 a2 b2 c2 d2 2.w 1/.a b c d/ 4w2 8w D 4 C C C C C C C 1 .4w2 4w 2.w 1/.a b c d/ D 4 C C C 1 w.w 1/ .w 1/.a b c d/ D C 2 C C C and so a C b C c C d w : D C 2 (17)

Thus we have the result. ut Example 6.4. For w D 9,wehaveseen,above4 f9I 5; 5; 3; 1I 11g sds and 4w D 36 D a2.D 12/ C b2.D 12/ C c2.D .3/2/ C d2.D 52/. 1 1 1 1 We have for w D 9, k1 D 2 .9 C1/, k2 D 2 .9 C1/, k3 D 2 .9 3/, k1 D 2 .9 C5/ 1 . / 9 1 .1 1 3 5/ 9 2 11 and D w C 2 a C b C c C d D C 2 C C D C D . Thus we have 4 f9I 5; 5; 3; 7I 11g sds. Remark 6.1. Lemma 6.1 actually applies to any four circulant matrices with elements of ˙1 which have row sums jaj; jbj; jcj; jdj which satisfy the additive property given in Eq. (2). On Good Matrices and Skew Hadamard Matrices 27

7 Conclusion

We have given for the ﬁrst time conditions to improve computer searches for good matrices and hence for skew-Hadamard matrices. Further research will be undertaken to implement this on various platforms.

References

1. Baumert, L.D.: Hadamard matrices of orders 116 and 232. Bull. Am. Math. Soc. 72, 237 (1966) 2. Baumert, L.D., Hall, Jr., M.: A new construction for Hadamard matrices. Bull. Am. Math. Soc. 71, 169–170 (1965) 3. Baumert, L.D., Hall, Jr., M.: Hadamard matrices of Williamson type. Math. Comp. 19, 442–447 (1965) 4. Baumert, L.D., Golomb, S.W., Hall, Jr., M.: Discovery of an Hadamard matrix of order 92. Bull. Am. Math. Soc. 68, 237–238 (1962) 5. -Dokovic,´ D.Z.: Construction of some new Hadamard matrices. Bull. Aust. Math. Soc. 45(2), 327–332 (1992) 6. -Dokovic,´ D.Z.: Ten new orders for Hadamard matrices of skew type. Elektrotehnickog Fak Ser. Matematika 3, 47–59 (1992) 7. -Dokovic,´ D.Z.: Good matrices of order 33, 35 and 127 exist. J. Comb. Math. Comb. Comput. 14, 145–152 (1993) 8. -Dokovic,´ D.Z.: Williamson matrices of order 4n for 33, 35 and 127. Discret. Math. 115, 267–271 (1993) 9. -Dokovic,´ D.Z.: Five new orders for Hadamard matrices of skew type. Australas. J. Comb. 10, 259–264 (1994) 10. -Dokovic,´ D.Z.: Supplementary difference sets with symmetry for Hadamard matrices (English summary). Oper. Matrices 3(4), 557–569 (2009) 11. -Dokovic,´ D.Z.: Regarding good matrices of order 41. Email communication to author, 23rd July 2014 12. Georgiou, S., Koukouvinos, C., Stylianou, S.: On good matrices, skew Hadamard matrices and optimal designs. Comput. Stat. Data Anal. 41(1), 171–184 (2002). ISSN 0167-9473 13. Hadamard, J.: Resolution d’une question relative aux determinants. Bull. Sci. Math. 17, 240–246 (1893) 14. Hall, Jr., M.: Combinatorial Theory, 2nd edn. Wiley, New York (1986) 15. Holzmann, W.H., Kharaghani, H., Tayfeh-Rezaie, B.: Williamson matrices up to order 59. Des. Codes Crypt. 46(3), 343–352 (2008) 16. Hunt, D.C.: Skew-Hadamard matrices of order less than 100. In: Wallis, J., Wallis, W.D. (eds.) Combinatorial Mathematics: Proceedings of the First Australian Conference, pp. 55–59. TUNRA, Newcastle (1971) 17. Hunt, D.C., Wallis, J.: Cyclotomy, Hadamard arrays and supplementary difference sets. Congr. Numer. 7, 351–382 (1972) 18. Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933) 19. Scarpis, V.: Sui determinanti di valoremassimo. Rend. R. Inst. Lombardo Sci. e Lett. 31(2), 1441–1446 (1898) 20. Seberry, J.: Good matrices online resource. http://www.uow.edu.au/~jennie/good.html (1999) 21. Seberry, J., Yamada, M.: Hadamard matrices, sequences, and block designs. In: Dinitz, J.H., Stinson, D.R. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 431–560. Wiley, New York (1992) 28 G. Awyzio and J. Seberry

22. Sylvester, J.J.: Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tesselated pavements in two or more colours, with applications to Newton’s rule, ornamental tile-work, and the theory of numbers. Phil. Mag. 34(4), 461–475 (1867) 23. Wallis, J.S.: A skew-Hadamard matrix of order 92. Bull. Aust. Math. Soc. 5, 203–204 (1971) 24. Wallis, J.S.: Combinatorial matrices. Ph.D. Thesis, La Trobe University, Melbourne (1971) 25. Wallis, J.S.: Hadamard matrices. In: Wallis, W.D., Street, A.P., Wallis, J.S. (eds.) Combina- torics: Room Squares, Sum-Free Sets and Hadamard Matrices. Lecture Notes in Mathematics. Springer, Berlin (1972) 26. Wallis, J.S.: Some remarks on supplementary sets. Inﬁnite and ﬁnite sets. Colloq. Math. Soc. Janos Bolyai 10, 1503–1506 (1973) 27. Wallis, J.S.: Williamson matrices of even order. In: Holton, D.A. (eds.) Combinatorial Mathe- matics: Proceedings of the Second Australian Conference. Lecture Notes in Mathematics, vol. 403, pp. 132–142. Springer, Berlin-Heidelberg-New York (1974) 28. Wallis, J.: Construction of Williamson type matrices. J. Linear Multilinear Algebra 3, 197–207 (1975) 29. Wallis, J., Whiteman, A.L.: Some classes of Hadamard matrices with constant diagonal. Bull. Aust. Math. Soc. 7, 233–249 (1972) 30. Williamson, J.: Hadamard’s determinant theorem and the sum of four squares. Duke Math. J. 11, 65–81 (1944) Suitable Permutations, Binary Covering Arrays, and Paley Matrices

Charles J. Colbourn

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract A set of permutations of length v is t-suitable if every element precedes every subset of t 1 others in at least one permutation. The maximum length of a t-suitable set of N permutations depends heavily on the relation between t and N. Two classical results, due to Dushnik and Spencer,p are revisited. Dushnik’s result determines the maximum length when t > 2N. On the other hand, when t is ﬁxed Spencer’s uses a strong connection with binary covering arrays of strength t 1 to obtain a lower bound on the length that is doubly exponential in t.We explore intermediate values for t, by ﬁrst considering directed packings and related Golomb rulers, and then by examining binary covering arrays whose number of rows is approximately equal to their number of columns. These in turn are constructed from Hadamard and Paley matrices, for which we present some computational data and questions.

Keywords Suitable sets of permutations • Hadamard matrix • Paley matrix • Golomb ruler • Directed block design

Mathematics Subject Classiﬁcation (2010): 05B20, 05B40, 05B05, 05A05, 06A07

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. C.J. Colbourn () School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, AZ 85287-8809, USA e-mail: [email protected]

© Springer International Publishing Switzerland 2015 29 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_3 30 C.J. Colbourn

1 Suitable Permutations and Suitable Cores

Hadamard matrices are pervasive. One of the many ways in which they have been used is in the construction of binary covering arrays of ‘large’ strength [4]. Binary covering arrays have in turn been used to determine lengths of t-suitable sets of N permutations [25]. We explore a set of questions that arise from these connections. We ﬁrst set the context for suitable sets of permutations, and then tour some perhaps surprising connections with combinatorial objects before focussing on Paley matrices. We consider a set P of permutations f1;:::;N g each on the set ˙ Df1;:::vg of v symbols. The set P is suitable of strength t,ort-suitable, if for every subset S Â ˙ of size t and every 2 S, there is a permutation 2 P for which 1./ < 1.s/ for every s 2 S nfg. Our interest is in SUN.t; N/ D maxfv W there exists a t-suitable set of N permutations of length vg. Forming an N v array A in which the entry in position .i; j/ is i.j/, one can equivalently say that for every set S of t symbols and every symbol 2 S,there is a row of A in which precedes all elements of S nfg.Thisisan.N;v;t/- suitable array. An equivalent formulation asks for an .N;v;t/-minrep array B,in which choosing any subset C of t columns, and any column 2 C,thereisarow in which the unique smallest element in the chosen columns in this row appears in column . The equivalence of .N;v;t/-suitable arrays and .N;v;t/-minrep arrays is immediate: Simply interchange the roles of symbols and columns. Here is a (7,9,4)- suitable array and the equivalent (7,9,4)-minrep array:

385762419 875962431 429675183 261895347 176598234 496783215 956287314 973681254 826745319 871645239 724536819 871423596 625389174 561482974

We state some easy observations, due to Dushnik [8]. An element of ˙ is a leader in P if, for some 2 P, .1/ D ( appears in the ﬁrst position). (In our example, the leaders are f1; 3; 4; 6; 7; 8; 9g.) Now suppose that 2 ˙ is a leader in a t-suitable set P with N permutations of length v. Choose one permutation 2 P for which .1/ D .Let 2 P, ¤ .If 1./ D j

Replacing by can only impact t-sets S Â ˙ that contain (the ﬁrst in each other set has not changed). But when 2 S, precedes all elements of S nfg in , and the change from to is inconsequential. This can be repeated until each leader appears ﬁrst exactly once, and no leader appears in the next v N positions. Applying this to our example, we obtain the .7;9;4/-suitable array on the left. On the right we have renamed symbols so that the leaders are f3; 4; 5; 6; 7; 8; 9g.

352678419 3 21678459 425769183 4 12769583 152896734 5 21896734 952687314 9 21687354 825476319 8 12476359 725436819 7 12436859 625389174 6 12389574

To find suitable sets with the fewest permutations, then, it suffices to consider the permutations with the leaders removed. A collection of N permutations of length v is a t-suitable core if it can be extended by the addition of N leaders, one for each permutation. Let SCN.t; N/ D maxfv W there exists a t-suitable core of N permutations of length vg.ThenSUN.t; N/ D SCN.t; N/ C N provided that N t. Hence we can focus on t-suitable cores and SCN. Lemma 1.1. Let P be a collection of N permutations of ˙ with j˙jDv 1.The following statements are equivalent: 1. P is a t-suitable core. 2. Whenever 1 Ä s Ä min.v; t/, in every set S Â ˙ of size s with 2 S, at least t C 1 s permutations must have precede all of S nfg. When these hold, P contains at least maxfs.t C 1 s/ W 1 Ä s Ä min.v; t/g permutations. Proof. We first show that (1) implies (2). Let Q be the t-suitable set for which P is the core. For every set S Dfa1;:::;asgÂ˙ of size 1 Ä s Ä min.v; t/,letPi Â P be the permutations in which ai precedes all of S nfaig.ThenPi \ Pj D;unless i D j. Moreover, Pi contains more than t s permutations: If it did not, then the set S, together with at most t s leaders, would not have ai first in any permutation of Q. Next we show that (2) implies (1). Let R be the permutations obtained from P, by adding a leader for each permutation (placing leaders in the final positions of the remaining permutations). Consider a set S of t symbols in R. Because v 1, 32 C.J. Colbourn

P contains at least t permutations, and hence when S contains only added leaders, every 2 S precedes all of Snfg in a permutation of R. So suppose that S contains t s leaders L and s 1 elements F of P.Every 2 L appears first among S in some permutation, so suppose that 2 F.Thens precedes all of F nfg in at least t C 1 s permutations, and since jLjDt s, there is a permutation of R in which precedes all of S nfg. When these hold, P must contain at least s.t C 1 s/ permutations for every 1 Ä s Ä min.v; t/. tC2 Lemma 1.2. When v Ä 2 , a t-suitable core with N permutations exists if and only if N v.t C 1 v/. Proof. That N v.t C 1 v/ is ensured by Lemma 1.1. It suffices to construct a t-suitable core on v symbols with v.t C1 v/ permutations. Place each symbol first in precisely t C 1 v permutations. Among each group of t C 1 v permutations, ensure that every other element appears second in at least one permutation. This can be done because t C 1 v v 1. Complete the permutations arbitrarily. Consider a set S of 1 Ä s Ä v elements and suppose that 2 S.Now precedes all of S nfg in the t C1 v permutations in which is placed first. Moreover, each of the v s groups of permutations in which no element of S is placed first contains a permutation in which is placed second. Hence precedes all of S nfg in at least t C 1 v C v s D t C 1 s permutations, so we have a t-suitable core. Removing any symbol from a t-suitable core yields a t-suitable core. So when v tC1 . tC1 /2 v t . t /. tC2 / 2 and t is odd, N 2 .Andwhen 2 and t is even, N 2 2 . Lemma 1.3. For s 1,thereisa2s-suitable core of length s C 2 if and only if N s.s C 1/. Proof. The lower bound follows by deleting one symbol and applying Lemma 1.1. To construct the suitable core, we use element set f1;:::;sg[fx; yg.Thereare s.s 1/ permutations (type O) in which the first two elements are .i; j/ for 1 Ä i; j Ä s, i ¤ j; s permutations (type X) in which the first three elements are .x; i:y/ for 1 Ä i Ä s;ands permutations (type Y) in which the first three elements are .y; i; x/ for 1 Ä i Ä s. Each of the type O permutations has third element x or y, subject to the rule that whenever .i; j; x/ starts a permutation, another is started by .j; i; y/. Now consider cases for a set S of size ` Ä s 1. Case 1. x 2 S and x is to precede S nfxg: This happens s times in X.Wheny 2 S,it s`C2 s`C2 2 1 ` also happens 2 times in O,ands C 2 s C .Wheny 62 S,it s`C1 s`C1 2 1 ` also happens 2 times in O,ands C 2 s C . Case 2. y 2 S and y is to precede S nfyg: This case is similar. Case 3. For some 1 Ä i Ä s, i 2 S and i is to precede S nfig:Thereares 1 permutations in O with i in the first position. Now if S \fx; ygDfx; yg,thereare at least s 1 .` 3/ permutations in O with i in the second position and no element of S in the first position; if S\fx; ygDfxg, there are at least s1.`2/ permutations in O with i in the second position and no element of S in the first position and also one permutation in Y;ifS \fx; ygDfyg, there are at least Suitable Permutations, Binary Covering Arrays, and Paley Matrices 33

s 1 .` 2/ permutations in O with i in the second position and no element of S in the first position and also one permutation in X;andifS \fx; ygD;, there are at least s 1 .` 1/ permutations in O with i in the second position andnoelementofS in the first position and also one permutation in X and one permutation in Y. In each case i precedes Snfig in at least 2sC1` permutations, as required. A more involved argument shows that the elements in a .s.sC1/; sC2;2s/-suitable core must be as in the proof of Lemma 1.3, and hence that SCN.2s; s.sC1// D sC2. When s D 2, a 4-suitable core of length 4 has permutations 21xy, 12yx, x2y1, x1y2, y2x1,andy1x2. It establishes that SUN.4; 6/ D 10, an improvement on our first example showing that SUN.4; 7/ 9. Lemma 1.4. For s 1,thereisa2s C 1-suitable core of length s C 2 if and only if N .s C 1/2. Proof. The lower bound follows by deleting a symbol and applying Lemma 1.1.To construct the suitable core, there are s.sC1/ permutations (type O)inwhichthefirst two elements are .i; j/ for 1 Ä i; j Ä s C 1, i ¤ j;ands C 1 permutations (type X) in which the first two elements are .s C 2;i/ for 1 Ä i Ä s C 1. The verification is straightforward. tC1 tC1 These results completely determine SCN.t; N/ when N < b 2 cd 2 e,and . ; tC1 tC1 / t 2 establish that SCN t b 2 cd 2 e b2 cC . A better lower bound on the number of permutations required when v>tC2 is needed to extend further. 2 p 2 1 In the cases for which exact results are obtained,p the strength is at least N , and the lengths (in the core) are bounded above by N C 2.

2 Directed Packings and Golomb Rulers

Füredi and Kahn [11] use an easy probabilistic argument to establish the general 2 statement that SUN.t; N/ t2N=t 1. Kierstead [20] focuses on cases when t is v approximately log , and obtains substantially better bounds for thesep situations. Here we consider the situation when the strength is approximately N in order to develop constructive methods. A 4-suitable core with seven permutations on seven symbols is provided by elements Z7, developing the first permutation 0,1,4,6,3,5,2 modulo 7. The sets that follow element 0 in the seven permutations are {1,4,6,3,5,2}; {3,5,2,4,1}; {2,6,1,5}; {4,6,3}; {2,6}; {4}; and ;. Then 0 appears before every 3- element subset once, every 2-element subset twice, and every element three times; by symmetry, every element satisfies the same conditions. So this forms a 4-suitable core. An easier explanation arises by considering the development of the initial 4- tuple .0;1;4;6/ modulo 7. The four 4-tuples containing 0 are (0,1,4,6), (6,0,3,5), (3,4,0,2), and (1,2,5,0). Any s-subset S f1; 2; 3; 4; 5; 6g with 0 Ä s Ä 3 can 34 C.J. Colbourn intersect at most s of the sets of elements preceding 0 (;, f6g, f3; 4g,andf1; 2; 5g), and hence S must follow 0 in at least 4 s permutations. Hence the development of any permutation with initial 4-tuple (0,1,4,6) yields a 4-suitable core. This example generalizes in a natural way. A k-tuple .a1; a2;:::;ak/ covers the (ordered) pairs f.ai; aj/ W 1 Ä i < j Ä kg.Ak-tuple is a k-block when all elements in the k-tuple are distinct. Choose a set T DfT1;:::;Tbg of k-blocks on a v-element set V.Forx 2 V, denote by rx the number of k-blocks of T that contain x.For x; y 2 V, denote by x;y the number of k-blocks of T that cover .x; y/.ThenT is a .v; b; k; t/-suitable packing ifP for every x 2 V and every S Â V nfxg with 0 1 ÄjSjDs Ä t ,wehaverx y2S y;x t s. Lemma 2.1. Let v; b; k; t be integers with v k t. If a .v; b; k; t/-suitable packing exists, a t-suitable core of length v with b permutations exists. Proof. Let T be a .v; b; k; t/-suitable packing. Extend each k-block arbitrarily to form a permutation. A directed .v; k;/-packing is a collection of k-blocks on v elements in which every ordered pair is covered by at most blocks. It is a directed .v; k;/-design, DD.v; k;/, when every ordered pair is covered by at most blocks. .v; ;1/ .v; 2v.v1/ ; ; 2.v1/ / Lemma 2.2. Every DD k is a k.k1/ k k1 -suitable packing. 2v.v1/ 2.v1/ Proof. The number of blocks is k.k1/ and rx D k1 D r for every element x. 2.v1/ Because r k by Fisher’s inequality, k1 s k s. We are interested in cases when the number of blocks is small, to minimize that the number of permutations. Hence the directed designs of most interest are the symmetric ones, having b D v and r D k. Our earlier example is a DD.7;4;1/ having seven blocks, with k D r D 4. However, very few such symmetric directed designs with D 1 are known. Indeed, omitting the ordering of the blocks, the design is a (symmetric) biplane, which are known only when k 2f3; 4; 5; 6; 9; 11; 13g [15]. We pursue a generalization. Let T be a k-tuple of integers .a1; a2;:::;ak/ with 0 Ä ai Ä ` for 1 Ä i Ä k,min.ai W 1 Ä i Ä k/ D 0,max.ai W 1 Ä i Ä k/ D `,and ai ¤ aj when i ¤ j. If all differences fai aj W i ¤ jg are distinct, T is a Golomb ruler of size k and length `. For example, (0,1,4,6) is a Golomb ruler of size 4 and length 6; the alert reader should notice that this sequence starts the first permutation in our example. An optimal Golomb ruler is one with minimum length for particular size k. Much is known about optimal Golomb rulers [6, 7], and the equivalent Sidon sets [9, 10] We relax the requirement on differences: When differences faj ai mod v W 1 Ä i < j Ä kg are all distinct, it is a directional modular Golomb ruler of size k and modulus v.(Thesearenot the same as the well-studied modular Golomb rulers in which all differences faj ai mod v W i ¤ jg are distinct. See [6].) Every Golomb ruler of length ` yields a directional modular one for each modulus v ` C 1, but the converse does not hold. It is known that the optimal Suitable Permutations, Binary Covering Arrays, and Paley Matrices 35

Golomb rulers of sizes 3, 4, 5, 6, 7, and 8 have lengths 3, 6, 11, 17, 25, and 34, respectively. (The ﬁrst two are starter blocks for a DD.4;3;1/ and a DD.7;4;1/.) Nevertheless, directional modular Golomb rulers exist as follows: 1. (0,1,6,3,10) for k D 5, v D 11 (a starter block for a DD.11; 5; 1/); 2. (0,1,3,23,7,17,12) for k D 7, v D 24; 3. (0,1,3,8,18,24,12) for k D 7, v D 25; 4. (0,1,5,7,18,27,30,15) for k D 8, v D 31; 5. (0,1,7,31,26,15,12,3) for k D 8, v D 32; 6. (0,1,3,12,31,25,18,8) for k D 8, v D 33;and 7. (0,1,3,9,26,33,19,14) for k D 8, v D 34. There is no directional modular Golomb ruler when v D 16 and k D 6;however, a DD.16; 6; 1/ does exist [1]. Lemma 2.3. If there is a directional modular Golomb ruler of size t and modulus v, there is a t-suitable core of length v.

Proof. Let .a1;:::;at/ be a directional modular Golomb ruler of modulus v.Forma set T of v t-blocks R0;:::;Rv1 by developing the elements modulo v. This forms a .v; t;1/-directed packing with v blocks, with rx D t for every element x. Hence it is a .v; v; t; t/-suitable packing. Apply Lemma 2.1 to produce the suitable core. Of course, asking for an optimal Golomb ruler, or even an optimal directional modular Golomb ruler, is more than is needed to construct a t-suitable core. One wants a directed packing that forms a suitable packing. Nevertheless, an old conjecture of Erdos˝ [9] can be restated as follows: For all k 3,thereisa Golomb ruler of size k whose length is at most k2; this has been veriﬁed for all k Ä 65;000 [6]. Assuming the truth of the conjecture of Erdos,˝ we conclude that SCN.t; t2/ t2. A more dramatic transition occurs as t is reduced further.

3 Binary Covering Arrays

We now turn to cases where t is at most log2.N/. Spencer [25] did the fundamental research. We require further deﬁnitions. Let N, k,and be positive integers. Let ˙ be an alphabet of v 1 symbols. Let A be an N k array with entries from ˙. Columns are factors;thelevels of factor i are the members of ˙.Let.c1;:::;c / be atupleof column indices (ci 2f1;:::;kg for 1 Ä i Ä ). Let . 1;:::; / be a -tuple with i 2 ˙ for 1 Ä i Ä ,and i D j whenever ci D cj. Then the -tuple f.ci; i/ W 1 Ä i Ä g is a -way interaction. Array A covers the -way interaction f.ci; i/ W 1 Ä i Ä g if, in at least one row of A ,theentryinrow and column ci is i for 1 Ä i Ä . Array A is a covering array CA.NI ;k;v//of strength when every -way interaction is covered in at least rows. Given parameters ;k;v;,the usual goal is to determine (or upper bound) CAN. ; k;v/, the minimum number of rows needed. Covering arrays with D 1 have been studied extensively (see [3], 36 C.J. Colbourn for example); in this case, the subscript is usually omitted. Our concern is with binary covering arrays (CA.NI ;k;2/s), a survey for which appears in [23]. We introduce (and slightly generalize) Spencer’s fundamental result: Theorem 3.1. Let k t 3 be integers. Suppose that a CA.NI t 1; k;2/exists. Then SCN.t; N/ 2k.

Proof. Let A D .aij/ be a CA.NI t1; k;2/on symbols {0,1} with columns indexed by C Df1;:::;kg. Treat columns as ordered by the usual ordering on integers. We k produce N linear orders f W 1 Ä Ä Ng of 2 elements, the subsets of C.To produce ,whenC1 and C2 are distinct subsets of C and is the smallest column index appearing in the symmetric difference .C1 [ C2/ n .C1 \ C2/,setC1 C2 when exactly one of the conditions 2 C1 and a D 1 holds; otherwise, C2 C1. Each so produced is a linear order, and hence a permutation. Now consider any s Ä t1 distinct C1;:::;Cs Â C and a further distinct D Â C. For 1 Ä i Ä s let i be the smallest integer in the symmetric difference of Ci and D. Then for 1 Ä i Ä s let i D 1 if i 2 D,and i D 0 otherwise. If in row we 1 1 find that a i D i for Ä i Ä s,thenD Ci for Ä i Ä s. Hence we need only establish that the s-way interaction f.i; i/ W 1 Ä i Ä sg is covered at least t s times in A .NowA is a CA.NI t 1; k;2/and hence also a CA2t1s .NI s; k;2/,and t1s 2 t s for 0 Ä s Ä t 1,soD precedes all of C1;:::;Cs Â C in at least t s permutations. Hence the N permutations form a t-suitable core. Spencer [25] establishes that this construction yields a t-suitable set of permutations; we give the stronger statement that it forms a t-suitable core, and hence we can lengthen by CAN.t 1; k;2/ further symbols. (This is inconsequential when k > CAN.t 1; k;2/, the ranges in which Spencer was most interested.) In terms of suitable sets, we have N Theorem 3.2. When t 3 is fixed, SUN.t; N/ is ˝.22 /. Proof. When 2 and v 2 are fixed, CAN. ; k;v/ is .log k/;thelower bound is easy counting, and the upper bound a simple probabilistic argument. Apply Theorem 3.1, with D t 1. Spencer proves the matching upper bound, so the maximum length is .22N /. Although employed primarily when t is fixed, Theorem 3.1 can be applied whenever a CA.NI t 1; k;2/ exists. While CAN.2; k;2/ is known precisely for all k [17, 22], CAN. ; k;2/ with 3 is not. In the application to suitable sets of permutations, a case of definite interest is when is “large” for a specific N. Evidently CAN. ; k;2/ 2 whenever k , because there are 2 -way interactions to cover in every -tuple of columns. Nevertheless we can use covering arrays to examine cases in which is .log n/. Most constructions for covering arrays have focused on strengths two through six, but see [16, 23, 26]forlarger strengths. Here we focus on one specific direction, the connection with Hadamard and Paley matrices. Suitable Permutations, Binary Covering Arrays, and Paley Matrices 37

4 Hadamard and Paley Matrices

Let q be an odd prime power. Form a q q matrix P D .pij/ with rows and columns indexed by the elements of Fq.Setpij D1 when i D j; pij DC1 when i ¤ j and i j is a square in Fq,andpij D1 when i ¤ j and i j is not a square. Then P is a Paley matrix Pq;see[24]. Using techniques from character theory, a strong existence result has been proved (in somewhat different vernacular, see [4]): 2 2t2 Theorem 4.1 ([13]). When q is a prime power and q > t 2 ,Pq is a CA.qI t; q;2/. Theorems 3.1 and 4.1 have an important corollary for suitable sets of permutations. Corollary 4.1. Let q be a prime power. Let t 3 be an integer. Then SCN.t; q/ 2q . 1/ < 1 whenever t C log2 t C 2 log2 q.

This contrasts with the situation when t is fixedp and lengths grow doubly exponentially; and with the situation when t is . N/ when lengths grow (at least) linearly. Here we apparently encounter a singly exponential growth in the lengths. Moreover, knowledge about the existence of specific binary covering arrays of ‘large’ strength can provide more precise bounds. Colbourn and Kéri [2, 4] establish that the bound in Theorem 4.1 on q can be reduced substantially when t is “small.” Paley matrices provide a primary means to construct Hadamard matrices. A Hadamard matrix of order n is an n n square matrix whose entries are either C1 or 1 and whose rows have dot product 0. A Paley Hadamard matrix is obtained from the Paley matrix Pq with q Á 3.mod 4/ by adding a headline and sideline of all C1s. We often write C for C1 and for 1. In [2, 4] various constructions of binary covering arrays from Hadamard matrices are developed. In order to generalize, we use ha1;:::;ati with ai 2f; Cg for 1 Ä i Ä t to denote f.a1;:::;at/; .a1;:::;at/g.Arowofa˙1 matrix A covers ha1;:::;ati in columns .1;:::;t/ when it contains one of the t-tuples in ha1;:::;ati on these columns. In every Hadamard matrix of order n 4, within any three columns, each of the four classes (hC; C; Ci, hC; C; i, hC; ; Ci,andhC; ; i) is covered in n exactly 4 rows in the chosen Á columns. Hence every Hadamard matrix A of order n A .2 3; ;2/ =4.2 3; ;2/ yields a CA nI n , A . But more is true: we obtain a CAn nI n . The type of a ˙1 matrix is the minimum number of times that any class ha1;:::;a4i is covered in any four columns [21]. Classification of Hadamard matrices by their type is used extensively in their enumeration [18, 19]. When a Hadamard matrix of order n and type 1 exists, we obtain a covering array of larger strength, a CA .2nI 4;n;2/. Deleting any 1 rows of the Hadamard matrix and then including all rows and their negations, a CA1.2n2. 1/I 4;n;2/ is formed. When the type of the Hadamard matrix is larger, the covering array has fewer rows. Hence finding Hadamard matrices of maximal type is consequential for covering arrays. 38 C.J. Colbourn

This admits a further set of generalizations. The t-type of a ˙1 matrix A is the t1 minimum number of times that any of the 2 classes ha1;:::;ati is covered in any n t columns of A.WhenA is Hadamard matrix of order n 4, its 3-type is 4 and its 4-type is its type. The t-type is not affected by multiplying any row or column by 1, not by column or row permutations. One could also consider coverage of tuples rather than classes; then column and row permutations are permitted, but multiplication by 1 is not. The t-halftype of A is the minimum number of times that any t-tuple .a1;:::;at/ is covered in any t columns. The t-halftype of A is at most half its t-type, but it can be much smaller. The importance of these notions is captured in the following: Lemma 4.1. Let A be an n m ˙1 matrix.

1. If A has t-type 1,aCA .2nI t; m;2/and a CA1.2n 2. 1/I t; m;2/both exist. 0 0 2. If A has t-halftype 1,aCA 0 .nI t; m;2/and a CA1.n C 1I t; m;2/both exist. Because these generalized types appear to be relevant to the construction of binary covering arrays, and hence to suitable sets of permutations, we computed the t-types and t-halftypes for the Paley matrices when q is a prime and q < 500. These are shown in Table 1; some interpretationÂ isÃ in order.Â Each tÃ-type (3t;4t;5t;6t) 1 1 1 is computed for four matrices Pq, 1 Pq , ,and For these values Â Ã Â Ã Pq 1 Pq 1 1 1 of q, the types of and are the same. Except in two cases shown Pq 1 Pq underlined, the types of Pq and 1 Pq are the same; in the underlined cases, the type of the ﬁrst matrix is oneÂ largerÃ than that of the second. Hence types are shown 1 1 in two columns, the ﬁrst for and the second for 1 Pq . 1 Pq Â Ã 1 For halftypes, no column 1 is added, so halftypes for the two matrices and Pq Pq are reported. Substantial computational effort has been invested in computing binary covering array numbers; see [27]. Nevertheless, the computation of types and halftypes of Paley matrices leads to slight improvements on some current world records, in particular CA.NI 5; k;2/for .N; k/ 2f.208; 104/, (380,379), (430,431), (462,463), (466,467), (486,487), (490,491), (498,499)g. It matches the best result from computational approaches when .N; k/ 2f.252; 128/; .260; 132/g Naturally Theorem 4.1 implies not only a lower bound on the strength of the covering array produced, but also of the t-type and t-halftype of the Paley matrix. However, the bound is by no means tight (at least for ‘small’ t) so one can hope for results that establish that all or some Paley matrices have generalized types larger than are guaranteed by Theorem 4.1. This has the potential to provide best known bounds on sizes of binary covering arrays having a number of rows linear in the Suitable Permutations, Binary Covering Arrays, and Paley Matrices 39

Table 1 Generalized types of Paley matrices q 3t 3ht 4t 4ht 5t q 3t 3ht 4t 4ht 5t 5ht 6t 7 2110 229 56 55 24 24 24 23 6665 11 321010 233 57 56 25 24 24 23 6665 13 2110 239 60 59 26 26 26 25 8887 17 3211 241 59 58 26 26 26 25 8887 19 542110 251 63 62 28 27 28 27 8887 23 652121 257 63 62 28 28 28 27 8887 29 652221 263 66 65 29 28 29 28 9898 31 873232 269 66 65 29 28 28 27 9987 37 873221 271 68 67 30 29 30 29 9998 41 983221 277 68 67 30 30 30 29 9998 43 11 10 4343 281 69 68 31 30 30 29 9998 47 12 11 4443 283 71 70 31 31 31 30 9898 53 12 11 4443 293 72 71 32 32 32 31 10 10 10 9 59 15 14 6565 307 77 76 34 34 34 33 11 11 11 10 61 14 13 5443 311 78 77 35 34 35 34 11 10 11 10 67 17 16 66651110 313 77 76 34 34 34 33 10 10 10 9 71 18 17 76761110 317 78 77 35 34 34 33 10 10 10 9 73 17 16 7665 331 83 82 37 36 37 36 11 11 11 10 79 20 19 87871110 337 83 82 37 36 36 35 11 11 11 10 83 21 20 88871110 347 87 86 39 38 39 38 13 13 13 12 89 21 20 88872221 349 86 85 39 38 38 37 12 12 12 11 97 23 22 98872221 353 87 86 39 38 38 37 12 12 12 11 101 24 23 10 10 10 9 2221 359 90 89 40 40 40 39 14 13 14 13 1110 103 26 25 10 10 10 9 1010 367 92 91 41 41 41 40 14 14 14 13 107 27 26 11 10 11 10 2221 373 92 91 41 40 40 39 14 14 14 13 109 26 25 11 10 10 9 2221 379 95 94 43 42 43 42 14 14 14 13 1010 113 27 26 11 10 10 9 2221 383 96 95 43 43 43 42 15 15 15 14 127 32 31 13 13 13 12 3332 389 96 95 43 43 42 41 14 14 14 13 131 33 32 14 13 14 13 3232 397 98 97 44 44 44 43 15 15 15 14 137 33 32 14 14 14 13 3332 401 99 98 45 44 44 43 14 13 13 12 139 35 34 15 14 15 14 3332 409 101 100 46 46 46 45 16 16 16 15 149 36 35 15 14 14 13 4443 419 105 104 47 47 47 46 16 16 16 15 151 38 37 16 15 16 15 4443 421 104 103 47 46 46 45 16 16 16 15 157 37 37 16 16 16 15 4443 431 108 107 49 48 49 48 16 15 16 15 2221 163 41 40 17 17 17 16 4443 433 107 106 49 48 48 47 16 16 16 15 10 167 42 41 18 17 18 17 4443 439 110 109 50 49 50 49 16 16 16 15 173 42 41 18 18 18 17 4443 443 111 110 50 50 50 49 17 17 17 16 179 45 44 19 19 19 18 5554 449 111 110 50 50 50 49 17 17 17 16 181 44 43 19 18 18 17 3221 457 113 112 51 50 50 49 18 18 18 17 191 48 47 21 20 21 20 6665 461 114 113 52 52 52 51 18 18 18 17 193 47 46 20 20 20 19 6665 463 116 115 53 52 53 52 18 18 18 17 2221 197 48 47 21 20 20 19 5554 467 117 116 53 53 53 52 19 18 19 18 2221 199 50 49 21 21 21 20 6565 479 120 119 55 54 55 54 19 19 19 18 211 53 52 23 22 23 22 6665 487 122 121 55 55 55 54 20 19 20 19 2221 223 56 55 24 24 24 23 7776 491 123 122 56 55 56 55 20 20 20 19 2221 227 57 56 25 24 25 24 7776 499 125 124 57 56 57 56 20 20 20 19 2221 40 C.J. Colbourn number of columns. In turn this can yield substantial improvements in constructions for suitable sets of permutations. Of particular interest to the Hadamard matrix community is the search for Hadamard matrices with large t-type or t-halftype.

5 Summary

To illustrate the approaches, consider strength t D 6. By Dushnik’s method and Lemma 1.3, 8 ˆ 0 when N <6 <ˆ 1 6 9 .6; / when Ä N Ä SCN N D ˆ 2 10 11 :ˆ when Ä N Ä 5 when N D 12

Lemmas 2.1 and 2.2 establish 8 < 7 when N D 14 .DD.7;3;1// .6; / 10 15 . .10; 4; 1// SCN N : when N D DD 16 when N D 16 .DD.16; 6; 1//

A simple backtracking method shows that SCN.6; 13/ 6 and SCN.6; 14/ 8; see Table 2. The Golomb ruler of length 17 ensures that SCN.6; N/ N for all N 18.

Table 2 (13,6,6)- and (14,6,8)-suitable arrays 014235 01352467 105234 12046357 215034 23165047 312045 34061257 420135 45210367 523014 56431027 025134 60251347 134025 07264135 243015 17402356 350124 27504136 451023 37652014 543012 47632015 034125 57310246 67105234 Suitable Permutations, Binary Covering Arrays, and Paley Matrices 41

Using Theorem 3.1 together with known covering arrays, we obtain 8 ˆ 64 when N D 32 .CA.32I 5; 6; 2/[14]/ ˆ ˆ 128 when N D 42 .CA.42I 5; 7; 2/[28]/ <ˆ 256 when N D 52 .CA.52I 5; 8; 2/[5]/ SCN.6; N/ ˆ 512 when N D 54 .CA.54I 5; 9; 2/[5]/ ˆ ˆ 1;024 when N D 56 .CA.56I 5; 10; 2/[5]/ :ˆ 16;384 when N D 64 .CA.64I 5; 14; 2/[5]/

Theorem 4.1 ensures that the Paley matrix Pq has nonzero 5-halftype when q > 6;400 is a prime power. However, the smallest Paley Hadamard matrix (when q is prime) with nonzero 5-type arises for q D 67, and the smallest Paley matrix with nonzero 5-halftype has q D 359.SoSCN.6; 136/ 295147905179352825856 D 268 and SCN.6; 359/ 2359. If we consider SCN.6; 132/, however, at present we can use a CA.132I 5; 24; 2/ [27] to get the bound SCN.6; 132/ 16777216 D 224, much lower than the bound for SCN.6; 136/. This gap suggests that there is much room for improvement, and in particular that constructions of ˙1 matrices of ‘large’ t-type or t-halftype have potential. We have seen that the study of suitable sets of permutations touches on many combinatorial objects, but we have only scratched the surface here. So let us be clear in closing that many other combinatorial objects may play a substantial role; for example, Kierstead’s work [20] establishes important connections with certain families of subsets with “small” intersections, and the work in [12] provides much more information when t D 3. Nevertheless, we expect that the connections discussed here can provide fruitful ways to explore the construction of suitable sets of permutations.

References

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7. Drakakis, K.: A review of the available construction methods for Golomb rulers. Adv. Math. Commun. 3(3), 235–250 (2009) 8. Dushnik, B.: Concerning a certain set of arrangements. Proc. Am. Math. Soc. 1, 788–796 (1950) 9. Erdos,˝ P.: On a problem of Sidon in additive number theory. Acta Sci. Math. Szeged 15, 255–259 (1954) 10. Erdos,˝ P., Turán, P.: On a problem of Sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. 16, 212–215 (1941) 11. Füredi, Z., Kahn, J.: On the dimensions of ordered sets of bounded degree. Order 3, 15–20 (1986) 12. Füredi, Z., Hajnal, P., Rödl, V., Trotter, W.T.: Interval orders and shift orders. In: Hajnal, A., Sos, V.T. (eds.) Sets, Graphs, and Numbers, pp. 297–313. North-Holland, New York/Amsterdam (1991) 13. Graham, R.L., Spencer, J.H.: A constructive solution to a tournament problem. Can. Math. Bull. 14, 45–48 (1971) 14. Hedayat, A.S., Sloane, N.J.A., Stufken, J.: Orthogonal Arrays. Springer, New York (1999) 15. Ionin, Y.J., Shrikhande, M.S.: Combinatorics of Symmetric Designs. New Mathematical Monographs, vol. 5. Cambridge University Press, Cambridge (2006) 16. Johnson, K.A., Entringer, R.: Largest induced subgraphs of the n-cube that contain no 4-cycles. J. Comb. Theory Ser. B 46, 346–355 (1989) 17. Katona, G.O.H.: Two applications (for search theory and truth functions) of Sperner type theorems. Period. Math. 3, 19–26 (1973) 18. Kharaghani, H., Tayfeh-Rezaie, B.: On the classiﬁcation of Hadamard matrices of order 32. J. Comb. Des. 18(5), 328–336 (2010) 19. Kharaghani, H., Tayfeh-Rezaie, B.: Hadamard matrices of order 32. J. Comb. Des. 21(5), 212–221 (2013) 20. Kierstead, H.A.: On the order dimension of 1-sets versus k-sets. J. Comb. Theory Ser. A 73, 219–228 (1996) 21. Kimura, H.: Classiﬁcation of Hadamard matrices of order 28. Discret. Math. 133(1–3), 171–180 (1994) 22. Kleitman, D., Spencer, J.: Families of k-independent sets. Discret. Math. 6, 255–262 (1973) 23. Lawrence, J., Kacker, R.N., Lei, Y., Kuhn, D.R., Forbes, M.: A survey of binary covering arrays. Electron. J. Comb. 18(1), Paper 84, 30 (2011) 24. Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933) 25. Spencer, J.: Minimal scrambling sets of simple orders. Acta Math. Acad. Sci. Hung. 22, 349–353 (1971/1972) 26. Tang, D.T., Chen, C.L.: Iterative exhaustive pattern generation for logic testing. IBM J. Res. Dev. 28, 212–219 (1984) 27. Torres-Jimenez, J., Rodriguez-Tello, E.: New upper bounds for binary covering arrays using simulated annealing. Inf. Sci. 185(1), 137–152 (2012) 28. Yan, J., Zhang, J.: A backtracking search tool for constructing combinatorial test suites. J. Syst. Softw. 81, 1681–1693 (2008) Divisible Design Digraphs

Dean Crnkovic´ and Hadi Kharaghani

Abstract Divisible design graphs (DDGs) have been recently deﬁned by Haemers, Kharaghani, and Meulenberg as a generalization of .v; k;/-graphs. In this paper we deﬁne and study divisible design digraphs (DDDs), a directed graph version of DDGs. On the other hand, DDDs are also natural generalization of doubly regular asymmetric digraphs. We obtain necessary conditions for the existence of a DDD with given parameters and give some constructions.

Keywords Directed graph • Divisible design • Divisible design graph

1 Introduction

Agraph can be interpreted as a design by taking the vertices of as points, and the neighborhoods of the vertices as blocks. Such a design is called a neighborhood design of . The adjacency matrix of is the incidence matrix of its neighborhood design. A k regular graph on v vertices with the property that any two distinct vertices have exactly common neighbors is called a .v; k;/-graph (see [14]). The neighborhood design of a .v; k;/-graph is a symmetric .v; k;/design. Haemers, Kharaghani, and Meulenberg have deﬁned divisible design graphs (DDGs for short) as a generalization of .v; k;/-graphs (see [7]). Further results on DDGs can be found in [6].

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. D. Crnkovic´ Department of Mathematics, University of Rijeka, 51000 Rijeka, Croatia e-mail: [email protected] H. Kharaghani () Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, Canada T1K3M4 e-mail: [email protected]

© Springer International Publishing Switzerland 2015 43 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_4 44 D. Crnkovic´ and H. Kharaghani

Definition 1.1. A k-regular graph is a divisible design graph (DDG for short) if the vertex set can be partitioned into m classes of size n, such that two distinct vertices from the same class have exactly 1 common neighbors, and two distinct vertices from different classes have exactly 2 common neighbors. In this paper we define and study DDDs, a directed graph version of DDGs. A directed graph (or digraph) is a pair D .V; E/,whereV is a finite nonempty set of vertices and E is a set of ordered pairs (arcs) .x; y/ with x; y 2 V and x ¤ y. Adigraph is asymmetric if .x; y/ 2 E implies .y; x/ … E.If.x; y/ is an arc, we will say that x dominates y or that y is dominated by x. A digraph is called regular of degree k if each vertex of dominates exactly k vertices and is dominated by exactly k vertices. We call a digraph on v vertices doubly regular with parameters .v; k;/ if it is regular of degree k and, for any distinct vertices x and y, the number of vertices z that dominates both x and y is equal to and the number of vertices z that are dominated by both x and y is equal to . For further information on doubly regular digraphs and doubly regular asymmetric digraphs, we refer the reader to [8–10]. Adigraph D .V; E/ on v vertices V Dfx1;:::;xvg may be characterized by its adjacency matrix, an v v .0; 1/-matrix A D Œaij defined by

aij D 1 if and only if .xi; xj/ 2 E:

An adjacency matrix of a doubly regular digraph is an incidence matrix of a symmetric design. Deﬁnition 1.2. Let be a regular asymmetric digraph of degree k on v vertices. is called a divisible design digraph (DDD for short) with parameters .v; k;1;2; m; n/ if the vertex set can be partitioned into m classes of size n,such that for any two distinct vertices x and y from the same class, the number of vertices z that dominates or being dominated by both x and y is equal to 1,andforany two distinct vertices x and y from different classes, the number of vertices z that dominates or being dominated by both x and y is equal to 2. DDDs are natural generalization of doubly regular asymmetric digraphs. Note that the adjacency matrix of a DDD with m D 1, n D 1,or1 D 2 is the incidence matrix of a symmetric design. In this case we call the DDD improper, otherwise it is proper. An incidence structure with v points and the constant block size k is a (group) divisible design with parameters .v; k;1;2; m; n/ whenever the point set can be partitioned into m classes of size n, such that two vertices from the same class have exactly 1 common neighbors, and two vertices from different classes have exactly 2 common neighbors. A divisible design D is said to be symmetric (or to have the dual property) if the dual of D is a divisible design with the same parameters as D. The deﬁnition of a DDD yields the following theorem.

Theorem 1.1. If is a DDD with parameters .v; k;1;2; m; n/, then its neighborhood design is a symmetric divisible design .v; k;1;2; m; n/. Divisible Design Digraphs 45

We say that a .0; 1/-matrix X is skew if X C Xt is a .0; 1/-matrix. Thus the adjacency matrix of a DDD is skew. If D is a symmetric divisible design .v; k;1;2; m; n/ that has a skew incidence matrix, then D is the neighborhood design of a DDD with parameters .v; k;1;2; m; n/. In this paper we obtain necessary conditions for the existence of a DDD with given parameters. Further, we present many constructions of such graphs.

2 The Quotient Matrix

Throughout the paper we denote by Iv, Ov,andJv the identity matrix, the zero- matrix, and the all-one matrix of size vv, respectively. Further, we denote by jv and ov the all-one column vector and the zero column vector of length v, respectively. Let us deﬁne K.m;n/ D Im ˝ Jn. Then the adjacency matrix A of a DDD with parameters .v; k;1;2; m; n/ is skew and satisﬁes:

t AA D kIv C 1.K.m;n/ Iv/ C 2.Jv K.m;n//: (1)

Moreover, if A is a skew matrix that satisﬁes Eq. (1), then A is an adjacency matrix of a DDD with parameters .v; k;1;2; m; n/. As pointed out in [7], taking row sums on both sides of the Eq. (1) yields

2 k D k C 1.n 1/ C 2n.m 1/:

So we have at most four independent parameters. Some obvious conditions are 1 Ä k Ä v 1, 0 Ä 1 Ä k,and0 Ä 2 Ä k 1. The vertex partition from the deﬁnition of a DDD gives a partition (called the canonical partition) of the adjacency matrix 2 3 A1;1 ::: A1;m 6 : : : 7 A D 4 : :: : 5 :

Am;1 ::: Am;m

Bose proved in [2] that the canonical partition of a symmetric divisible design is a tactical decomposition of its incidence matrix A, which means that each block Ai;j has constant row and column sum. That enables us to deﬁne the matrix R D Œri;j, where ri;j is the row (and column) sum of Ai;j. The matrix R is called the quotient matrix of A. 46 D. Crnkovic´ and H. Kharaghani

The following theorem was proved in [2] for symmetric divisible designs, and much shorter proof is given in [7]: Theorem 2.1. Let R be the quotient matrix of a proper DDD with parameters .v; k;1;2; m; n/. Then R satisﬁes

t 2 RR D .k 2v/Im C 2nJm:

The following theorem follows directly from the deﬁnition of a DDD.

Theorem 2.2. Let R D Œrij be the quotient matrix of a proper DDD˘ with .v; ; ; ; ; / n1 parameters k 1 2 m n .Thenrij C rji Ä nwheni¤ j, and rii Ä 2 . Proof. This is a direct consequence of the fact that the adjacency matrix A is skew, i.e. A C At is a .0; 1/-matrix.

Corollary 2.1. If is a proper DDD with parameters .v; k;1;2; m; n/,then n 1 .m 1/n k : Ä 2 C 2

Theorem 2.3 gives a simple observation about an automorphism of a DDD with parameters .v; k;1;2; m;2/.

Theorem 2.3. If is a proper DDD with parameters .v; k;1;2; m;2/,then v admits an automorphism of order two acting in 2 orbits of length two. v v Proof. The canonical partition divides the adjacency matrix in 2 2 blocks of dimension 2 2. There are four possibilities for these 2 2 blocks: Ä Ä Ä Ä 00 01 10 11 ; ; : 00 10 01 and 11

Every block matrix consisting of these blocks admits an automorphism of order two permuting the rows and columns of each block.

3 Nonexistence Results

In this section we establish nonexistence of some DDDs whose parameters survive the conditions given in Sect. 2. 2 It follows from the equation k D k C 1.n 1/ C 2n.m 1/ that for a proper DDD.v; k;1;2; m; n/ such that 2n D k the number of vertices k is greater than or equal to m. Especially, if a DDD is proper, 2n D k and 1 D 0,thenk D m. Further, the following theorem holds.

Theorem 3.1. There is no DDD.v; k;0;2; m; n/ such that k > m. Divisible Design Digraphs 47

Proof. Let R be the quotient matrix of a DDD.v; k;0;2; m; n/.Ifk > m, then there exists an entry rij of R such that rij 2. Then two blocks of the divisible design which belong to the same class intersect in one or more points, that contradicts the fact that 1 D 0. Let a and b be real numbers. Then 0 Ä .a b/2, and consequently 2ab Ä a2 C 2 ; Rm . ;:::; / . ;:::; / Pb . Similarly, if x Py 2 ,whereP x D x1 xm and y D y1 ym ,then m m 2; m 2 iD1 xiyi Ä maxPf iD1 xi iDP1 yi g. Especially, if is a permutation of the set 1;:::; m m 2 f mg,then iD1 xix.i/ Ä iD1 xi . We use these facts to prove the following theorem. 2 Theorem 3.2. There is no DDD.v; k;1;2; m; n/ such that k <2v.

Proof. Let D be a DDD.v; k;1;2; m; n/ and let R be its quotient matrix. Then t 2 RR D .k 2v/Im C 2nJm is a positive semi-deﬁnite matrix with an eigenvalue 2 k 2v.

Parameter sets .v; k;1;2; m; n/ for DDDs that survive the conditions given in Sect. 2 are called feasible. Theorem 3.3. The following feasible DDDs do not exist: .12; 5; 0; 2; 6; 2/ .18; 8; 4; 3; 3; 6/ .22; 9; 5; 2; 2; 11/ .25; 8; 4; 2; 5; 5/ .14; 4; 0; 1; 7; 2/ .18; 7; 3; 2; 2; 9/ .24; 8; 7; 2; 8; 3/ .26; 11; 7; 2; 2; 13/ .15; 6; 3; 2; 5; 3/ .20; 9; 0; 4; 10; 2/ .24; 10; 6; 2; 2; 12/ .27; 10; 9; 3; 9; 3/ .15; 5; 4; 1; 5; 3/ .20; 9; 8; 3; 5; 4/ .24; 8; 4; 1; 2; 12/ .27; 6; 3; 1; 9; 3/ .16; 7; 6; 2; 4; 4/ .20; 8; 4; 2; 2; 10/ .25; 12; 8; 5; 5; 5/ .27; 11; 7; 3; 3; 9/ .18; 5; 4; 1; 9; 2/ .21; 9; 5; 3; 3; 7/ .25; 9; 8; 2; 5; 5/ .27; 7; 3; 1; 3; 9/ .18; 7; 6; 2; 6; 3/

Proof. We will explicitly give a proof of nonexistence just for a few parameter sets, the rest of the cases can be handled in a similar way. The matrix

M D J6 I6 is the only quotient matrix of a DDD.12; 5; 0; 2; 6; 2/. By performing the exhaustive search we found out that there is no DDD.12; 5; 0; 2; 6; 2// with the quotient matrix M. Conducting the exhaustive search we came to conclusion that there is no DDD with parameters .14; 4; 0; 1; 7; 2/. We have used the fact that the quotient matrix of a DDD.14; 4; 0; 1; 7; 2/ is the incidence matrix of a symmetric (7,3,1) design. If R is the quotient matrix of a DDD.15; 6; 3; 2; 5; 3/,thenRj5 D 6j5 and t RR D 6I5 C 6J5. It follows that the entries of a row of R take values from the multiset S1 Df3; 1; 1; 1; 0g or S2 Df2;2;2;0;0g. Two rows with entries from the multiset S1 cannot have inner product equal to 6, and the same holds for the multiset S2. Since one cannot construct a quotient matrix R with required properties, a DDD.15; 6; 3; 2; 5; 3/ does not exist. 48 D. Crnkovic´ and H. Kharaghani

The quotient matrix R of a DDD.18; 7; 6; 2; 6; 3/ satisﬁes conditions Rj6 D 7j6 t and RR D 13I6 C 6J6. The entries of a row of the matrix R must take values from the multiset S Df3; 3; 1; 0; 0; 0g.SinceRt is also the quotient matrix of a DDD.18; 7; 6; 2; 6; 3/, each column of R has the entries from the multiset S.That contradicts the fact that the inner product of any two distinct rows of R is equal to 6. Using similar arguments one can prove the nonexistence of DDDs in the remaining cases.

4 Construction of DDDs

In this section we present constructions of DDDs known to us.

4.1 Construction from Symmetric Designs

Lemma 4.1. Let D be a skew incidence matrix of a symmetric .v; k;/design and P be a permutation matrix of size t t. Then the Kronecker product P ˝ Disthe adjacency matrix of a DDD with parameters .vt; k;;0;t;v/. Lemma 4.2. Let D be a skew incidence matrix of a symmetric design with parameters .v; k;/.ThenD˝Jn is the adjacency matrix of a DDD with parameters .vn; kn; kn;n;v;n/. Lemma 4.3. Let M be the core of a normalized symmetric Hadamard matrix of order 4n and N the core of a normalized skew-type Hadamard matrix of order 4m. C C Split M D M M , and N I4m1 D N N into positive and negative parts. Let D D MC ˝ NC C M ˝ N,thenDC Dt is a .0; 1/-matrix. Furthermore,

t DD D.4nm 3n m C 1/J4n1 ˝ J4m1 C .n m/J4n1 ˝ I4m1 (2) nI4n1 ˝ J4m1 C n.4m 1/I4n1 ˝ I4m1:

C Proof. Note that M C M D J4n1, M M D nJ4n1 C nI4n1,andJ4n1M D C C C M J4n1. It follows that M M D M M D nJ4n1 nI4n1.SinceN C N D C t C C J4m1 I4m1, .N / D N , we conclude that N N C N N D .2m 1/J4m1 t C C .2m 1/I4m1. It is now easy to see that D C D D M ˝ .N C N / C M ˝ .NC C N/ is a .0; 1/-matrix and the Eq. (2) is valid. Remark. Let D0 D MC ˝ NC C M ˝ NC, then by a symmetry argument D0 is also a DDD with the same parameters, so we have a twin. Note that Dt D D0. Theorem 4.1. Let D be a symmetric incidence matrix of a Hadamard .4l C 3; 2l C 1; l/ design. Further, let D1 be a skew incidence matrix of a Hadamard .4l C 3; 2l C 1; l/ design and D1 D J4lC3 D1 be the incidence matrix of its complementary Divisible Design Digraphs 49 design. Replace each entry value 1 of the matrix D by D1, and each entry value 0 of DbyD1 I4lC3. The resulting matrix M is the adjacency matrix of a DDD..4l C 3/2;.4l C 3/.2l C 1/; l.4l C 3/; .2l C 1/2;4l C 3; 4l C 3/. Proof. The statement follows directly from Lemma 4.3,bytakingn D m D l C 1. Corollary 4.1. Let q be a prime power, q Á 3.mod 4/. Then there exists a DDD 2 q1 q3 q1 2 with parameters .q ; q 2 ; q 4 ;. 2 / ; q; q/. . / . 1/ Proof. The q q matrix D1 D dij , such that 1 1; if .j i/ is a nonzero square in GF.q/; d D ij 0; otherwise;

. ; q1 ; q3 / is an incidence matrix of a Hadamard q 2 4 design. Such a symmetric design is called the a Paley design (see [13]). Since 1 is not a square in GF.q/, D1 is askewmatrix.LetD D D1R,whereR is the back diagonal matrix, then D is a . ; q1 ; q3 / symmetric matrix and an incidence matrix of a Hadamard q 2 4 design. Theorem 4.2. Let there be a Hadamard .4l C 3; 2l C 1; l/ design with a skew incidence matrix, such that 4l C 5 is a prime power. Then there exists a DDD..4l C 5/.4l C 3/; .4l C 4/.2l C 1/; l.4l C 4/; .2l C 1/2;4l C 5; 4l C 3/. Proof. Let q D 4l C 5 be a prime power. Then q Á 1.mod 4/ and a .q q/ matrix C D .cij/ deﬁned as follows: 1; if .j i/ is a nonzero square in GF.q/; c D ij 0; otherwise; is a symmetric matrix, since 1 is a square in GF.q/. There are as many nonzero . / q1 squares as nonsquares in GF q , so each row of C has 2 elements equal 1 and qC1 2 zeros. The set of nonzero squares in GF.q/ is a partial difference set, called a Paley partial difference set (see [1, 10.15 Example, pp. 231]), and the matrix C is the adjacency matrix of the Paley graph (see, e.g., [3]and[15, Section 3.1]). Let C D Œcij be .q q/ matrix such that cij D cij C 1.mod 2/. Further, let Ci and Cj, i ¤ j,betheith and the jth row of the matrix C, respectively. It is known (see [5]) that q1 t 4 ; if cij D cji D 0; Ci Cj D q1 1; 1: 4 if cij D cji D

Further, if Ci and Cj, i ¤ j,aretheith and the jth row of the matrix C D .cij/ respectively, then q1 t 4 ; if cij D cji D 0; Ci Cj D q1 4 C 1; if cij D cji D 1: 50 D. Crnkovic´ and H. Kharaghani

Therefore, the row products of the matrix .C Iq/ are the same as in the case of the matrix C,and 0; ; . /t if i D j Ci C Iq j D q1 4 ; otherwise:

Let D be a skew incidence matrix of a Hadamard .4l C 3; 2l C 1; l/ design. Then D D J4lC3 D is an incidence matrix of a Hadamard .4l C 3; 2l C 2;l C 1/ design, and D I4lC3 is a skew incidence matrix of a Hadamard .4l C 3; 2l C 1; l/ design. The matrix C ˝ D C .C Iq/ ˝ .D I4lC3/ is the adjacency matrix of a DDD..4l C 5/.4l C 3/; .4l C 4/.2l C 1/; l.4l C 4/; .2l C 1/2;4l C 5; 4l C 3/. Corollary 4.2. There exists a DDD with parameters (15,4,0,1,5,3). Proof. The existence of a DDD.15; 4; 0; 1; 5; 3/ follows from Theorem 4.2,forl D 0. The matrices C and D from the proof of Theorem 4.2 are given as follows: 2 3 01001 6 7 2 3 6 101007 010 6 7 6 010107 ; 4 0015 : C D 6 7 D D 4 001015 100 10010

Lemma 4.4. Let there be a Hadamard .4l C 3; 2l C 1; l/ design with a skew incidence matrix. Then there exists a DDD.28lC21; 8lC7; 4lC3; 2lC2;7;4lC3/. Proof. Let D be the incidence matrix of the Hadamard (7,3,1) design: 2 3 0110100 6 7 6 00110107 6 7 6 00011017 6 7 6 10001107 ; D D 6 7 6 01000117 6 7 4 10100015 1101000

D1 be a skew incidence matrix of a Hadamard .4l C 3; 2l C 1; l/ design, and D1 D J4lC3 D1. Then the matrix D˝D1 CI7 ˝D1 is the adjacency matrix of a DDD.28lC 21; 8l C 7; 4l C 3; 2l C 2;7;4l C 3/. Lemma 4.5. Let D be a skew incidence matrix of a Hadamard .4l C 3; 2l C 1; l/ design, and D1 be a skew incidence matrix of a symmetric .v1; k1;1/ design. Replace each diagonal entry of the matrix D by D1, each off-diagonal entry value 0 of D by Ov1 and each entry value 1 of D by Jv1 . The resulting matrix is the adjacency matrix of a DDD..4l C 3/v1;.2l C 1/v1 C k1;.2l C 1/v1 C 1;v1l C k1;4l C 3; v1/. Divisible Design Digraphs 51

Remark. Another way to describe the adjacency matrix of a DDD from Lemma 4.5 is D˝Jv1 CIv ˝D1. Note that D and D1 can be incidence matrices of trivial designs. For example, if both symmetric designs have parameters (3,1,0), the constructed DDD has parameters (9,4,3,1,3,3). We found useful to use circulant block matrix D1 D circ.0;1;0;:::;0/.

4.2 Construction from Hadamard Matrices

Let H be a regular Hadamard matrix of order 4u2 with row sum 2u. Replace each entry value 1 of H by 0, then we obtain the incidence matrix of a symmetric .4u2;2u2 C u; u2 C u/ design, called a Menon design. Note that the integer u can be positive or negative. The complementary design of a Menon design with parameters .4u2;2u2 C u; u2 C u/ is a Menon design with parameters .4u2;2u2 u; u2 u/.A Hadamard matrix H is skew if H C Ht D 2I. Theorem 4.3. Let H be a skew Hadamard matrix of order 4u with diagonal entries equal to 1. Then there exists a DDD with parameters .8u;4u 1; 0; 2u 1; 4u;2/.

Proof. Replace each diagonal entry of H by O2, each entry value 1 of H by I2,and each entry value 1 by J2 I2. The obtained matrix is the adjacency matrix of a DDD with parameters .8u;4u 1; 0; 2u 1; 4u;2/. Jennifer Seberry had conjectured that a skew Hadamard matrix of order n exists if and only if n D 1; 2 or 4u. This conjecture is conﬁrmed for n < 188 (see [4]). For example, a DDD(8,3,0,1,4,2) can be constructed using the skew Hadamard matrix 2 3 1111 6 7 6 1111 7 H D 4 5 : 1 111 1111

Theorem 4.4. If there exist a Menon design with parameters .4u2;2u2 u; u2 u/ that has a skew incidence matrix D and a Hadamard matrix of order 2u2, then there exists a DDD with parameters .24u2;12u2 2u;4u2 2u;6u2 2u;12u2;2/. Proof. Let D be the incidence matrix of a Menon .4u2;2u2 u; u2 u/ design. Suppose that D is a skew matrix. Further, let H1 and H2 be Hadamard matrices 2 2 of order 4u and 2u , respectively. Note that H1 does not have to be related to the Menon design with the incidence matrix D. DeﬁneamatrixD1 D D ˝ J2. Further, denote by H3 the matrix obtained from H1 in such a way that we replace each 1 by ŒI2jI2, and each 1 by ŒJ2 I2jJ2 I2. Let us deﬁne auxiliary matrices u and v as follows: 52 D. Crnkovic´ and H. Kharaghani

2 3 2 3 0110 1001 6 10017 6 01107 6 7 ;v6 7 ; u D 4 10015 D 4 01105 0110 1001 and the matrix H4 such that each entry of H2 with value 1 is replaced by u, and each 1 is replaced by v. If X is the incidence matrix of an incidence structure I, let us denote by X the incidence matrix of the complementary structure of I. Deﬁne the matrix B as follows: " # D1 H4 B D t : H4 D1

One can verify that the matrix " # D1 H3 A D t H3 B is a skew matrix that satisﬁes

t .12 2 2 / .4 2 2 /. / .6 2 2 /. /; AA D u u I24u2 C u u K.12u2;2/ I24u2 C u u J24u2 K.12u2;2/ where K.m;n/ D Im ˝ Jn. That means that A is the adjacency matrix of a DDD with parameters .24u2;12u2 2u;4u2 2u;6u2 2u;12u2;2/. Corollary 4.3. There exists a DDD with parameters (24,10,2,4,12,2). Proof. Applying the construction from Theorem 4.4, with matrices 2 3 2 3 0010 1111 Ä 6 7 6 7 6 00017 6 111 1 7 11 D D 4 5 ; H1 D 4 5 ; H2 D : 0100 1 111 1 1 1000 1 1 11

We now introduce DDDs that can be constructed from skew balanced generalized weighing matrices. The main tools for the construction methods consist of the following: • Skew balanced generalized weighing matrices with zero diagonal over a variety of cyclic groups [8]. • Block nega-circulant Bush-type Hadamard matrices [11, 12].

Let ck D circ.0;1;0;:::;0/ be the shift circulant matrix of order k with the ﬁrst row .0;1;0;:::;0/ and let nk D negacirc.0;1;0;:::;0/be the nega-circulant Divisible Design Digraphs 53 matrix of order k with the ﬁrst row .0;1;0;:::;0/. The cyclic group generated by ck and nk are of order k and 2k, respectively. We denote these groups by Ck and N2k. Theorem 4.5. Suppose that 4h2 is the order of a block nega-circulant Bush-type Hadamard matrix. Let ` be a positive integer, such that q D 4h.2` 1/ C 1 is a prime power. Let m D 1 C q C q2 CCq2sC1, s a positive integer. Then there is a DDD with parameters

.4h2m;.2h2 h/q2sC1;.h2 h/q2sC1;.2` 1/.2h 1/.2h2 h/q2s; m;4h2/:

Proof. Let H be a block nega-circulant Bush-type Hadamard matrix of order 4h2 and let W D Œwij be a skew balanced generalized weighing matrix with parameters .1 C q C q2 CCq2mC1; q2mC1k0; q2m.q 1// with zero diagonal over the cyclic group G generated by N2h ˝ I2h. We ﬁrst replace the diagonal blocks of H with the zero blocks. We get a twin symmetric .4h2;2h2 h; h2 h/ design, see [8]. Denote C the twin design by D.LetL D R2h ˝I2h, and deﬁne Q D ŒMwijL.LetQ D Q Q , then each of QC and Q are DDD with parameters

.4h2m;.2h2 h/q2sC1;.h2 h/q2sC1;.2` 1/.2h 1/.2h2 h/q2s; m;4h2/:

2s The pair of DDDs above is called a twin DDD. Noting that 21 D .2`h/hq , all DDDs obtained from this theorem are proper except when h D 2`. Corollary 4.4. If 2h is the order of a Hadamard matrix and q D 4h.2` 1/ C 1 is a prime power, then there exist a DDG with the parameters of Theorem 4.5. Proof. In this case, existence of the required nega-circulant Bush-type Hadamard matrix follows from the construction in [12] with some obvious modiﬁcation. As an example, for h D ` D m D 1, we have a twin DDD with parameters .24; 5; 0; 1; 6; 4/ from Theorem 4.5, as follows. Example 4.1. Let 2 3 044444 6 7 62034127 6 7 62103247 6 7 ; W D 62210437 6 7 42342015 242130 be a skew BGW.6;5;4/ overÄ the cyclic group GÄ of order four generatedÄ by the 0 I2 0 C 1 block nega-circulant matrix .LetH D ,whereC D ,be I2 0 C 0 1 the trivial block nega-circulant twin design obtained from the Bush-type Hadamard matrix of order 4. Applying the technique from Theorem 4.5 we get a twin DDD with parameters .24; 5; 0; 1; 6; 4/. 54 D. Crnkovic´ and H. Kharaghani

There are only two known block nega-circulant Bush-type Hadamard matrix of order 4h2, h odd, the trivial one of order 4 and the non-trivial of order 36, see [11].

4.3 Miscellaneous Constructions

In this subsection we describe constructions of DDDs that do not use Hadamard matrices or symmetric designs. Theorem 4.6. For every odd integer n, n 3, there exists a DDD with parameters .4n; n C 2;n 2;2;4;n/.

Proof. Let Cn D circ.0;1;0;:::;0/be a .n n/ matrix and let In D Jn In.Then 2 3 Cn In Cn I 6 7 6 ICn In Cn 7 H1 D 4 5 Cn ICn In In Cn ICn is the adjacency matrix of a DDD with parameters .4n; n C 2;n 2;2;4;n/. Theorem 4.7. Let p be an odd prime. Then there exists a DDD with parameters .p2; p;0;1;p; p/. 2 2 Proof. First we construct the .p p / matrix M D ŒmkpCr;lpCs,wherek; l; r; s D 0;:::;p 1, such that 1; if s Á kl C r .mod p/; m ; D kpCr lpCs 0; otherwise:

Let us prove that the matrix M is the incidence matrix of a symmetric divisible design with parameters .p2; p;0;1;p; p/.Thep classes of the divisible design are determined by k D 0; 1; : : : ; p1. It is clear that the scalar product of two rows from the same class, the rows kp C r1 and kp C r2, is zero. Since the equation k1l C r1 D 1 k2l C r2 in GF.p/ has exactly one solution for l, namely l D .r2 r1/.k1 k2/ ,the scalar product of two rows from different classes, the rows k1p C r1 and k2p C r2,is one. Divisible Design Digraphs 55

The matrix M is not the adjacency matrix of a DDD.p2; p;0;1;p; p/, because its first k rows have diagonal entries equal to one. Define the .p2 p2/ matrix A D ŒakpCr;lpCs, k; l; r; s D 0;:::;p 1, such that mkpCrC1;lpCs; if k D 0; akpCr;lpCs D mkpCr;lpCs; otherwise; where the index kp C r C 1 is calculated in GF.p/. The matrix A is skew, so it is the adjacency matrix of a DDD.p2; p;0;1;p; p/. Remark. If B is the incidence matrix of a symmetric divisible design with parameters .n2; n;0;1;n; n/, then the following matrix 2 3 t t 1 j o 2 6 p p 7 4 t 5 jp Op Ip ˝ jp op2 Ip ˝ jp B is the incidence matrix of a projective plane of order n (provided that the rows and the columns of B are ordered with respect to the classes of the divisible design and its dual). Therefore, the matrices M and A from the proof of Theorem 4.7 can be used for construction of a projective plane of order p. Below we describe constructions of DDDs with parameter sets (12,4,2,1,6,2), (16,7,2,3,4,4), (16,4,0,1,4,4), (18,6,0,2,6,3), and (28,6,2,1,7,4). We did not find a way to generalize these constructions, so we call them sporadic. Theorem 4.8. There exists a DDD with parameters (12,4,2,1,6,2). Proof. The matrix 2 3 O2 J2 I2 I2 O2 O2 6 7 6 O2 O2 J2 O2 I2 I2 7 6 7 6 7 6 O2 O2 O2 J2 I2 I2 7 ; 6 7 6 I2 I2 O2 O2 J2 O2 7 4 I2 I2 O2 O2 O2 J2 5 J2 O2 I2 I2 O2 O2 where I2 D J2 I2, is the adjacency matrix of a DDD.12;4;2;1;6;2/. Theorem 4.9. There exists a DDD with parameters (16,7,2,3,4,4). Proof. The following matrix 56 D. Crnkovic´ and H. Kharaghani 2 3 0100110010101001 6 7 6 00100110010111007 6 7 6 00010011101001107 6 7 6 10001001010100117 6 7 6 7 6 01100001001110107 6 7 6 00111000100101017 6 7 6 10010100110010107 6 7 6 11000010011001017 6 7 6 01011001010011007 6 7 6 7 6 10101100001001107 6 7 6 01010110000100117 6 7 6 10100011100010017 6 7 6 00110101011000017 6 7 6 10011010001110007 6 7 4 11000101100101005 0110101011000010 is the adjacency matrix of a DDD with parameters (16,7,2,3,4,4). Theorem 4.10. There exists a DDD with parameters (16,4,0,1,4,4). Proof. Let us define matrices U, V, W,andZ as follows: 2 3 2 3 0010 0001 6 00017 6 00107 6 7 ; 6 7 ; U D 4 01005 V D 4 10005 1000 0100 2 3 2 3 0100 1000 6 10007 6 01007 6 7 ; 6 7 : W D 4 00105 Z D 4 00015 0001 0010

The matrix 2 3 UUUU 6 7 6 UVWZ7 4 5 UZVW UW Z V is the adjacency matrix of a DDD(16,4,0,1,4,4). Divisible Design Digraphs 57

Theorem 4.11. There exists a DDD with parameters (18,6,0,2,6,3). Proof. We deﬁne the matrices X and Y as follows: 2 3 2 3 010 001 X D 4 0015 ; Y D 4 1005 : 100 010

The matrix 2 3 XXYI3 YX 6 7 6 XXXYI3 Y 7 6 7 6 7 6 YXXXYI3 7 6 7 6 XYI3 YYI3 7 4 YXI3 YXX5 XYXXXX is the adjacency matrix of a DDD(18,6,0,2,6,3). Theorem 4.12. There exists a DDD with parameters (28,6,2,1,7,4).

Proof. Let us deﬁne auxiliary matrices u1, u2,andu3 as follows: 2 3 2 3 2 3 1100 1010 1001 6 11007 6 01017 6 01107 6 7 ; 6 7 ; 6 7 : u1 D 4 00115 u2 D 4 10105 u3 D 4 01105 0011 0101 1001

The circulant block matrix D D circ.0; u1; u2;0;u3;0;0/is the adjacency matrix of a DDD with parameters (28,6,2,1,7,4). Note that the quotient matrix of a DDD(28,6,2,1,7,4) from Theorem 4.12 is equal 2D,whereD is the incidence matrix of a Hadamard (7,3,1) design obtained by the Paley construction. In Tables 1 and 2 we give all putative parameter sets .v; k;1;2; m; n/ for DDDs on at most 27 vertices that survive the conditions given in Sect. 2. We have omitted the cases when 1 D k or 2 D 0.ExamplesofDDDs with 1 D k or 2 D 0 are given in Theorems 4.1 and 4.2. The tables give 94 parameter sets. For each parameter set we tried to decide on existence or nonexistence. In 20 cases we do not know the answer.

Acknowledgements This work was started while Crnkovic´ was visiting University of Lethbridge. He is grateful for the support and hospitality provided by University of Lethbridge. Crnkovic´ also acknowledges the support of the Croatian Science Foundation grant 1637. Kharaghani thanks NSERC for the continuing support of his research. 58 D. Crnkovic´ and H. Kharaghani

Table 1 Feasible parameters of proper DDDs with v Ä 21, 0<2 < k, 1 < k

v k 1 2 mn Existence Reference 830142Yes Theorem 4.3 943133Yes Lemma 4.5 940233No Theorem 3.1 930133Yes Corollary 4.1, Theorem 4.7 12 5 1 2 4 3 Yes Theorem 4.6 12 5 4 1 3 4 Yes Lemma 4.5 12 5 0 2 6 2 No Theorem 3.3 12 4 2 1 6 2 Yes Theorem 4.8 12 4 0 2 2 6 No Theorem 3.1 12 3 0 1 2 6 No Theorem 3.1 14 4 0 1 7 2 No Theorem 3.3 14 5 1 2 2 7 No Theorem 3.2 15 6 3 2 5 3 No Theorem 3.3 15 5 4 1 5 3 No Theorem 3.3 15 4 0 1 5 3 Yes Corollary 4.2 15 6 0 3 3 5 No Theorem 3.1 15 6 5 1 3 5 Yes Lemma 4.5 15 5 0 2 3 5 No Theorem 3.1 16 7 0 3 8 2 Yes Theorem 4.3 16 7 2 3 4 4 Yes Theorem 4.9 16 7 6 2 4 4 No Theorem 3.3 16 4 0 1 4 4 Yes Theorem 4.10 18 5 4 1 9 2 No Theorem 3.3 18 7 6 2 6 3 No Theorem 3.3 18 6 0 2 6 3 Yes Theorem 4.11 18 8 4 3 3 6 No Theorem 3.3 18 7 6 1 3 6 Yes Lemma 4.5 18 4 0 1 3 6 No Theorem 3.1 18 7 3 2 2 9 No Theorem 3.3 20 9 0 4 10 2 No Theorem 3.3 20 8 2 3 10 2 ?– 20 7 6 2 10 2 ?– 20 5 2 1 10 2 ?– 20 9 8 3 5 4 No Theorem 3.3 20 9 3 4 4 5 ?– 20 7 3 2 4 5 Yes Theorem 4.6 20 6 0 2 4 5 No Theorem 3.1 20 8 4 2 2 10 No Theorem 3.3 20 6 0 3 2 10 No Theorem 3.1 20 5 0 2 2 10 No Theorem 3.1 21 10 0 5 7 3 No Theorem 3.1 (continued) Divisible Design Digraphs 59

Table 1 (continued)

v k 1 2 mn Existence Reference 21 10 9 4 7 3 Yes Lemma 4.5 21 9 0 4 7 3 No Theorem 3.1 21 8 1 3 7 3 ?– 21 7 3 2 7 3 Yes Theorem 4.4 21 10 1 6 3 7 No Theorem 3.2 21 10 8 3 3 7 Yes Lemma 4.5 21 9 5 3 3 7 No Theorem 3.3 21 8 0 4 3 7 No Theorem 3.1 21 8 7 1 3 7 Yes Lemma 4.5 21 7 0 3 3 7 No Theorem 3.1

Table 2 Feasible parameters v k 1 2 mn Existence Reference of proper DDDs with 22501112?– 22 Ä v Ä 27, 0<2 < k, 1 < k 22 9 5 2 2 11 No Theorem 3.3 24 11 0 5 12 2 Yes Theorem 4.3 24 10 2 4 12 2 Yes Corollary 4.3 24963122?– 24 10 3 4 8 3 ?– 24 8 7 2 8 3 No Theorem 3.3 24 7 0 2 8 3 ?– 24 11 10 4 6 4 ?– 24 9 4 3 6 4 ?– 24 5 0 1 6 4 Yes Theorem 4.5 24 11 4 5 4 6 ?– 24 10 0 5 4 6 No Theorem 3.1 24 9 0 4 4 6 No Theorem 3.1 24 8 4 2 4 6 ?– 24 11 2 6 3 8 No Theorem 3.2 24 10 6 3 3 8 ?– 24 9 8 1 3 8 Yes Lemma 4.5 24 6 2 1 3 8 ?– 24 10 6 2 2 12 No Theorem 3.3 24 9 0 6 2 12 No Theorem 3.1 24 8 4 1 2 12 No Theorem 3.3 24 4 0 1 2 12 No Theorem 3.1 25 12 3 6 5 5 No Theorem 3.2 25 12 8 5 5 5 No Theorem 3.3 25 9 8 2 5 5 No Theorem 3.3 25 8 4 2 5 5 No Theorem 3.3 25 5 0 1 5 5 Yes Theorem 4.7 (continued) 60 D. Crnkovic´ and H. Kharaghani

Table 2 (continued) v k 1 2 mn Existence Reference 26903132?– 26 11 7 2 2 13 No Theorem 3.3 26 10 1 6 2 13 No Theorem 3.2 27 12 6 5 9 3 ?– 27 11 7 4 9 3 ?– 27 10 9 3 9 3 No Theorem 3.3 2790393?– 2784293?– 2763193No Theorem 3.3 27 12 3 6 3 9 No Theorem 3.2 27 11 7 3 3 9 No Theorem 3.3 27 10 0 5 3 9 No Theorem 3.1 27 10 9 1 3 9 Yes Lemma 4.5 2790439No Theorem 3.1 2773139No Theorem 3.3

References

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Dean Crnkovic,´ Sanja Rukavina, and Vladimir D. Tonchev

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract The binary codes spanned by the blocks of the six known symmetric (61,16,4) designs and their automorphism groups are used for the construction of 17,350 new nonisomorphic symmetric designs with these parameters.

Keywords Symmetric design • Linear code • Automorphism group

1 Introduction

The terminology and notation for designs and codes in this paper follow [1, 3, 7]. We use the notation .v; k;/for a symmetric 2-.v; k;/design. According to the table of 2-.v; k;/designs of small order given in the Handbook of Combinatorial Designs [7, p. 39], there are at least six known nonisomorphic symmetric (61,16,4) designs (cf. [5, 8, 9, 11, 13, 14]). A design with these parameters seems to appear for the ﬁrst time in a paper by Mitchell [11]. In [5], Cepuli´ c´ proved that up to isomorphism, there exists exactly one symmetric (61,16,4) design admitting an automorphism of order 15 acting with orbits of length 1 or 15. We checked by computer that this design constructed by Cepuli´ c´ is isomorphic to the design listed in Trung’s chapter on symmetric designs [14, p. 82], as well as to the (61,16,4) design listed in Hall’s book [8, p. 413–414] (a note of caution:

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. D. Crnkovic´ • S. Rukavina Department of Mathematics, University of Rijeka, 51000 Rijeka, Croatia e-mail: [email protected]; [email protected] V.D. Tonchev () Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA e-mail: [email protected]

© Springer International Publishing Switzerland 2015 61 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_5 62 D. Crnkovicetal.´ the generating permutation listed on page 414 of Hall’s book [8] contains some typographical errors: the cycles starting with 4 and 49 should be (4,17,22,31,44) and (49,50,61,60,52), respectively). The design of Cepuli´ c´ is self-dual and has full automorphism group of order 270. In [9], Landjev and Topalova found five symmetric (61,16,4) designs that admit a nonabelian automorphism group of order 10. One of these designs is self-dual and isomorphic to the design constructed by Cepuli´ c,´ and the other four designs are two pairs of dual designs having full automorphism groups of order 90 or 30, respectively. The full automorphism group of Mitchell’s design is of order 27. In [6] Cepuli´ c´ showed that there are exactly five symmetric (61,16,4) designs that admit an automorphism of order 15. These designs are isomorphic to the designs found by Landjev and Topalova. In the sequel, we will denote Mitchell’s design by D0, and the designs constructed by Cepuli´ c,´ Landjev, and Topalova by D1;:::;D5, following the notation from [9]. This paper summarizes the results of a search for symmetric (61,16,4) designs in the binary linear codes C0;:::;C5 of length 61 spanned by the incidence vectors of blocks of the designs D0;:::;D5, respectively. This approach was used before in [12] to find a new quasi-symmetric 2-.56; 16; 6/ design in the binary code spanned by the incidence matrix of a known design with these parameters. Using Magma [4] we examined the sets of codewords of weight 16 to find subcollections of 61 codewords that compose incidence matrices of symmetric (61,16,4) designs. Since a complete search seemed infeasible, we restricted our search to designs that are invariant under a code automorphism of order 3 having one fixed point, and we were able to find all such designs, up to isomorphism. There are 8,536 such symmetric (61,16,4) designs in the codes of D0;:::;D5,which together with their duals form a set of 17,058 nonisomorphic designs. We also found 148 symmetric (61,16,4) designs invariant under an automorphism group of order 3 that has 16 fixed points. None of these 148 designs is self-dual, and together with their duals give 296 nonisomorphic designs. Thus, we obtained a total of 17,354 nonisomorphic symmetric (61,16,4) designs, 17,350 of which have not been previously known.

2 Designs from the Code of D0

In 1979, Mitchell [11] found an inﬁnite class of symmetric designs with parameters 2-.qhC1 q C 1; qh; qhl/,whereh 2 is an arbitrary integer, and q >2is a prime power such that there is an afﬁne plane of order q 1 (hence, q 1 is also a prime power). This construction uses divisible designs and results from [2]. Choosing q D 4 and h D 2, Mitchell’s construction gives a self-dual symmetric (61,16,4) design that we denote by D0. The full automorphism group of D0 has order 27 and structure E9 W Z3. Below we give the incidence vectors of the blocks of D0, which span the code C0. 1111000000000111000000000111000000000111000000000111000000000 1111000000000000111000000000111000000000111000000000111000000 New Symmetric (61,16,4) Designs Obtained from Codes 63

1111000000000000000111000000000111000000000111000000000111000 1111000000000000000000111000000000111000000000111000000000111 1000111000000111000000000000111000000000000111000000000000111 1000111000000000111000000111000000000000000000111000000111000 1000111000000000000111000000000000111111000000000000111000000 1000111000000000000000111000000111000000111000000111000000000 1000000111000111000000000000000111000000000000111000111000000 1000000111000000111000000000000000111000000111000111000000000 1000000111000000000111000111000000000000111000000000000000111 1000000111000000000000111000111000000111000000000000000111000 1000000000111111000000000000000000111000111000000000000111000 1000000000111000111000000000000111000111000000000000000000111 1000000000111000000111000000111000000000000000111111000000000 1000000000111000000000111111000000000000000111000000111000000 0100100100100100100100100100100100100100100100100000000000000 0100010010010100010010010100010010010100010010010000000000000 0100001001001100001001001100001001001100001001001000000000000 0010100001010010100001010010100001010010100001010000000000000 0010010100001010010100001010010100001010010100001000000000000 0010001010100010001010100010001010100010001010100000000000000 0001100010001001100010001001100010001001100010001000000000000 0001010001100001010001100001010001100001010001100000000000000 0001001100010001001100010001001100010001001100010000000000000 0100100100100000000000000001001001001010010010010100100100100 0100010010010000000000000001100100100010001001001100010010010 0100001001001000000000000001010010010010100100100100001001001 0010100001010000000000000100001010100001010100001010100001010 0010010100001000000000000100100001010001001010100010010100001 0010001010100000000000000100010100001001100001010010001010100 0001100010001000000000000010001100010100010001100001100010001 0001010001100000000000000010100010001100001100010001010001100 0001001100010000000000000010010001100100100010001001001100010 0000000000000100100100100010010010010001001001001100100100100 0000000000000100010010010010001001001001100100100100010010010 0000000000000100001001001010100100100001010010010100001001001 0000000000000010100001010001010100001100001010100010100001010 0000000000000010010100001001001010100100100001010010010100001 0000000000000010001010100001100001010100010100001010001010100 0000000000000001100010001100010001100010001100010001100010001 0000000000000001010001100100001100010010100010001001010001100 0000000000000001001100010100100010001010010001100001001100010 0001001001001010010010010100100100100000000000000100100100100 0001100100100010001001001100010010010000000000000100010010010 0001010010010010100100100100001001001000000000000100001001001 0100001010100001010100001010100001010000000000000010100001010 64 D. Crnkovicetal.´

0100100001010001001010100010010100001000000000000010010100001 0100010100001001100001010010001010100000000000000010001010100 0010001100010100010001100001100010001000000000000001100010001 0010100010001100001100010001010001100000000000000001010001100 0010010001100100100010001001001100010000000000000001001100010 0010010010010001001001001000000000000100100100100100100100100 0010001001001001100100100000000000000100010010010100010010010 0010100100100001010010010000000000000100001001001100001001001 0001010100001100001010100000000000000010100001010010100001010 0001001010100100100001010000000000000010010100001010010100001 0001100001010100010100001000000000000010001010100010001010100 0100010001100010001100010000000000000001100010001001100010001 0100001100010010100010001000000000000001010001100001010001100 0100100010001010010001100000000000000001001100010001001100010

The binary linear code C0 spanned by the blocks of D0 is a self-orthogonal doubly even [61,25,8] code with the automorphism group of order 31,104. The automorphism group of C0 has 18 conjugacy classes of subgroups of order 3; six classes having one fixed point, six classes having 16 fixed points, four classes having 31 fixed points, and two classes with 46 fixed points. It follows from [10, Corollary .v; ;/ 3.7] that if a nonidentity automorphismp of a nontrivial symmetric k design fixes f points, then f Ä k C k . Therefore, if an automorphism of order 3 of a symmetric (61,16,4) design fixes f points, then f Ä 19. Our aim was to find symmetric (61,16,4) designs that admit an automorphism of order 3 fixing one point, whose blocks are codewords of the code C0. Generators of representatives of the six conjugacy classes of subgroups of order 3 that have one fixed point are:

1. 0;1 D .1; 27; 26/.2; 50; 39/.3; 14; 52/.4; 40; 16/..5; 8; 11/.6; 9; 12/ .7; 10; 13/.15; 38; 51/.17; 20; 23/.18; 21; 24/.19; 22; 25/.29; 32; 35/.30; 33; 36/ .31; 34; 37/.41; 44; 47/.42; 45; 48/.43; 46; 49/.53; 56; 59/.54; 57; 60/.55; 58; 61/, 2. 0;2 D .2; 3; 4/.5; 7; 6/.8; 10; 9/.11; 13; 12/.14; 15; 16/.17; 19; 18/, .20; 22; 21/.23; 25; 24/.26; 27; 28/.29; 31; 30/.32; 34; 33/.35; 37; 36/.38; 39; 40/ .41; 43; 42/.44; 46; 45/.47; 49; 48/.50; 51; 52/.53; 55; 54/.56; 58; 57/.59; 61; 60/, 3. 0;3 D .2; 4; 3/.5; 7; 6/.8; 10; 9/.11; 13; 12/.14; 16; 15/.17; 19; 18/ .20; 22; 21/.23; 25; 24/.26; 28; 27/.29; 31; 30/.32; 34; 33/.35; 37; 36/.38; 40; 39/ .41; 43; 42/.44; 46; 45/.47; 49; 48/.50; 52; 51/.53; 55; 54/.56; 58; 57/.59; 61; 60/, 4. 0;4 D .2; 3; 4/.5; 11; 8/.6; 12; 9/.7; 13; 10/.14; 15; 16/.17; 23; 20/ .18; 24; 21/.19; 25; 22/.26; 27; 28/.29; 35; 32/.30; 36; 33/.31; 37; 34/.38; 39; 40/ .41; 47; 44/.42; 48; 45/.43; 49; 46/.50; 51; 52/.53; 59; 56/.54; 60; 57/.55; 61; 58/, 5. 0;5 D .2; 3; 4/.5; 7; 6/.8; 10; 9/.11; 12; 13/.14; 15; 16/.17; 19; 18/ .20; 22; 21/.23; 24; 25/.26; 27; 28/.29; 31; 30/.32; 34; 33/.35; 36; 37/.38; 39; 40/ .41; 43; 42/.44; 46; 45/.47; 48; 49/.50; 51; 52/.53; 55; 54/.56; 58; 57/.59; 60; 61/, 6. 0;6 D .2; 4; 3/.5; 7; 6/.8; 10; 9/.11; 12; 13/.14; 16; 15/.17; 19; 18/ .20; 22; 21/.23; 24; 25/.26; 28; 27/.29; 31; 30/.32; 34; 33/.35; 36; 37/.38; 40; 39/ .41; 43; 42/.44; 46; 45/.47; 48; 49/.50; 52; 51/.53; 55; 54/.56; 58; 57/.59; 60; 61/. New Symmetric (61,16,4) Designs Obtained from Codes 65

The permutation 0;1 acts on a symmetric (61,16,4) design with one fixed point (and one fixed block), and 20 orbits of length 3. C0 has 20,239 codewords of weight 16, and 0;1 partitions these codewords into 6,741 orbits of length 3, and fixes 16 codewords. Among the 6,741 orbits of length 3 there are 4,404 “good” orbits, e.g., orbits consisting of 3 codewords that pairwise have dot product equal to four. These “good” orbits together with a codeword fixed by 0;1 form 279,936 symmetric (61,16,4) designs. Among the 279,936 designs obtained, there are 4,230 mutually nonisomorphic ones. In the set of these 4,230 mutually nonisomorphic symmetric (61,16,4) designs there are four self-dual designs (one of them is isomorphic to D0) and one pair of mutually dual designs, hence the designs invariant under the action of 0;1 together with their dual designs produce the set of 8,454 mutually nonisomorphic symmetric (61,16,4) designs. Three self-dual designs have isomorphic full automorphism groups with the structure E9 W Z3, and the fourth self-dual design has the full automorphism group of order 108, with the structure .E9 W Z3/ E4. The pairwise dual designs have full automorphism group of order 108, with the structure .E9 W Z3/ E4. The permutation 0;2 acts on the codewords of weight 16 with 6,741 orbits of length 3, fixing 16 codewords. Among these 6,741 orbits, there are 4,608 “good” orbits. There are no more than 8 “good” orbits that can be put together so that all codewords have pairwise dot product 4, hence one cannot build a symmetric (61,16,4) design out of h 0;2i-orbits. The same results are obtained with the orbits of the permutations 0;3, 0;5 and 0;6. The permutation 0;4 also acts on the codewords of weight 16 in 6,741 orbits of length 3, fixing 16 codewords. Among the 6,741 orbits of length 3 there are 4,404 “good” orbits, which together with a fixed codeword produce 279,936 symmetric (61,16,4) designs. In this set of 279,936 designs there are 4,230 mutually nonisomorphic ones. Among them there are four self-dual designs and one pair of mutually dual designs, and those six designs are isomorphic to the designs obtained from the orbits of h 0;1i. Hence the designs invariant under the action of 0;4 together with their duals produce the set of 8,454 mutually nonisomorphic symmetric (61,16,4) designs. The designs constructed from the orbits of h 0;1i and h 0;4i together with their duals give us 16,860 mutually nonisomorphic symmetric (61,16,4) designs.

3 Designs from the Codes of D1, D3, D4 and D5

The designs D1, D3, D4 and D5 span isomorphic codes. D4 is the dual design of D1, and the full automorphism groups of these two designs have order 90 and structure E9 D10. D5 is the dual design of the design D2 with the full automorphism group of order 30 isomorphic to Z3 D10. D3 is a self-dual design with full automorphism group of order 270 and the structure .E9 W Z3/ D10. We searched for designs in the code C3 of the self-dual design D3. The binary linear code C3 spanned by the blocks of D3 is a self-orthogonal doubly even [61,25,8] code with the automorphism group 66 D. Crnkovicetal.´ of order 77,760. As in the case of the code C0, the automorphism group of C3 has 18 conjugacy classes of subgroups of order 3; six classes having one fixed point, six classes having 16 fixed points, four classes having 31 fixed points, and two classes with 46 fixed points. Our aim was to find symmetric (61,16,4) designs that admit an automorphism of order 3 that fixes one point, whose blocks are codewords of the code C3. Generators of representatives of the six conjugacy classes of subgroups of order 3 having one fixed point are:

1. 3;1 D .1; 9; 14/.2; 10; 11/.3; 15; 12/.5; 13; 16/.6; 8; 7/.17; 32; 47/ .18; 33; 48/.19; 34; 49/.20; 35; 50/.21; 36; 51/.22; 42; 57/.23; 43; 58/.24; 44; 59/ .25; 45; 60/.26; 46; 61/.27; 37; 52/.28; 38; 53/.29; 39; 54/.30; 40; 55/.31; 41; 56/, 2. 3;2 D .2; 12; 7/.3; 13; 8/.4; 14; 9/.5; 15; 10/.6; 16; 11/.17; 22; 27/ .18; 23; 28/.19; 24; 29/.20; 25; 30/.21; 26; 31/.32; 42; 37/.33; 43; 38/.34; 44; 39/ .35; 45; 40/.36; 46; 41/.47; 52; 57/.48; 53; 58/.49; 54; 59/.50; 55; 60/.51; 56; 61/; 3. 3;3 D .2; 12; 7/.3; 13; 8/.4; 14; 9/.5; 15; 10/.6; 16; 11/.17; 27; 22/ .18; 28; 23/.19; 29; 24/.20; 30; 25/.21; 31; 26/.32; 37; 42/.33; 38; 43/.34; 39; 44/ .35; 4045/.36; 41; 46/.47; 57; 52/.48; 58; 53/.49; 59; 54/.50; 60; 55/.51; 61; 56/; 4. 3;4 D .2; 12; 7/.3; 13; 8/.4; 14; 9/.5; 15; 10/.6; 16; 11/.17; 27; 22/ .18; 28; 23/.19; 29; 24/.20; 30; 25/.21; 31; 26/.32; 37; 42/.33; 38; 43/.34; 39; 44/ .35; 40; 45/.36; 41; 46/.47; 52; 57/.48; 53; 58/.49; 54; 59/.50; 55; 60/.51; 56; 61/; 5. 3;5 D .2; 12; 7/.3; 13; 8/.4; 14; 9/.5; 15; 10/.6; 16; 11/.17; 22; 27/ .18; 23; 28/.19; 24; 29/.20; 25; 30/.21; 26; 31/.32; 42; 37/.33; 43; 38/.34; 44; 39/ .35; 45; 40/.36; 46; 41/.47; 57; 52/.48; 58; 53/.49; 59; 54/.50; 60; 55/.51; 61; 56/, 6. 3;6 D .2; 12; 7/.3; 13; 8/.4; 14; 9/.5; 15; 10/.6; 16; 11/.17; 47; 32/ .18; 48; 33/.19; 49; 34/.20; 50; 35/.21; 51; 36/.22; 57; 42/.23; 58; 43/.24; 59; 44/ .25; 60; 45/.26; 61; 46/.27; 52; 37/.28; 53; 38/.29; 54; 39/.30; 55; 40/.31; 56; 41/.

The permutation 3;1 acts on a symmetric (61,16,4) design with one fixed point (and block) and 20 orbits of length 3. The code C3 has 27,466 codewords of weight 16, and 3;1 acts on these codewords with 9,150 orbits of length 3, fixing 16 codewords. Among the 9,150 orbits of length 3 there are 5,340 “good” orbits, e.g. orbits consisting of 3 codewords that pairwise have dot product equal to four. These “good” orbits together with a codeword fixed by 3;1 form 7,776 symmetric (61,16,4) designs. Among the 7,776 designs there are 58 mutually nonisomorphic ones. Two of these 58 designs are self-dual, the design D3 and a design with the full automorphism group of order 54 and the structure .E9 W Z3/ Z2. Among the other 56 designs there are no pairs of mutually dual designs. Therefore, these 58 designs and their duals give us 114 mutually nonisomorphic designs. The permutation 3;2 acts on the codewords with 9,150 orbits of length 3, fixing 16 codewords. Among the 9,150 orbits of length 3 there are 6,525 “good” orbits. There is no more than 8 “good” orbits, i.e., 24 codewords that can be put together so that all codewords have pairwise dot product 4, therefore one cannot build a symmetric (61,16,4) design out of h 3;2i-orbits. We obtained the same results with orbits of the permutations 3;3, 3;4 and 3;5. The permutation 3;6 also has 16 fixed codewords and 9,150 orbits of length 3. There are 5,340 good orbits of length 3, that together with a fixed codeword build New Symmetric (61,16,4) Designs Obtained from Codes 67

7,776 symmetric (61,16,4) designs, 58 of them mutually nonisomorphic. Among these 58 nonisomorphic designs two designs are self-dual, isomorphic to the self- dual designs invariant under the action of 3;1. These 58 designs and their duals produce 114 mutually nonisomorphic designs. The designs invariant under 3;1 together with the designs invariant under 3;6, and their duals, form a set of 204 mutually nonisomorphic designs. The designs constructed from C0 and C3, together with their duals, give 17,058 nonisomorphic symmetric (61,16,4) designs.

4 Designs from the Code of D2

D2 is a symmetric (61,16,4) design with the full automorphism group of order 30, isomorphic to Z3 D10. The binary linear code C2 spanned by the blocks of D2 is a self-orthogonal doubly even [61,25,8] code with full automorphism group of order 180 and structure .Z15 Z3/ W Z4. The automorphism group of C2 has four conjugacy classes of subgroups of order 3, one with 46 fixed points, one with 16 and two with one fixed point. The permutation that has 46 fixed points cannot act as an automorphism of a symmetric (61,16,4) design. Among the codewords of C2, we searched for designs that admit the automorphism group generated by the permutation

2;1 D .2; 7; 12/.3; 8; 13/.4; 9; 14/.5; 10; 15/.6; 11; 16/.17; 22; 27/

.18; 23; 28/.19; 24; 29/.20; 25; 30/.21; 26; 31/.32; 37; 42/.33; 38; 43/.34; 39; 44/

.35; 40; 45/.36; 41; 46/.47; 52; 57/.48; 53; 58/.49; 54; 59/.50; 55; 60/.51; 56; 61/; having one fixed point. The code C2 has 24,331 codewords of weight 16. The automorphism 2;1 acts on these codewords with 8,105 orbits of length 3, fixing 16 codewords. Among the 8,105 orbits of length 3 there are 4,355 “good” orbits. These “good” orbits do not produce a symmetric (61,16,4) design, because 15 is the maximal number of orbits that can be put together so that codewords have pairwise dot product 4. The same results are obtained using the permutation 2;2 defined as follows:

2;2 D .2; 7; 12/.3; 8; 13/.4; 9; 14/.5; 10; 15/.6; 11; 16/.17; 22; 27/

.18; 23; 28/.19; 24; 29/.20; 25; 30/.21; 26; 31/.32; 42; 37/.33; 43; 38/.34; 44; 39/

.35; 45; 40/.36; 46; 41/.47; 52; 57/.48; 53; 58/.49; 54; 59/.50; 55; 60/.51; 56; 61/; that also ﬁxes one point. 68 D. Crnkovicetal.´

The automorphism group generated by

2;3 D .2; 7; 12/.3; 8; 13/.4; 9; 14/.5; 10; 15/.6; 11; 16/.17; 22; 27/

.18; 23; 28/.19; 24; 29/.20; 25; 30/.21; 26; 31/.47; 52; 57/

.48; 53; 58/.49; 54; 59/.50; 55; 60/.51; 56; 61/ has 16 fixed points. The automorphism 2;3 acts on these codewords with 151 fixed codewords and 8,060 orbits of length 3. Among the 8,060 orbits of length 3 there are 1,160 “good” orbits. From these 1,160 orbits one gets 7,776 partial 2-(61,16,4) designs, i.e., sets of 15 orbits that form 45 blocks that pairwise have dot product equal to 4. In the set of 151 fixed codewords there is only one subset of 16 codewords that mutually intersect in 4 coordinate positions. Therefore, fixed blocks of the symmetric (61,16,4) designs that are constructed as unions of h 2;3i-orbits are uniquely determined. All of the 7,776 partial designs are compatible with the 16 fixed blocks, i.e., together with these 16 blocks they form 7,776 symmetric (61,16,4) designs. Among the constructed 7,776 designs there are 148 nonisomorphic ones, which are not self-dual. These designs and their duals give us 296 mutually nonisomorphic symmetric (61,16,4) designs, that are not isomorphic to any of designs constructed from C0 and C3.

5 Conclusion

We construct new symmetric (61,16,4) designs from the sets of codewords of weight 16 in the binary codes of the six known symmetric (61,16,4) designs. We find all designs invariant under an automorphism of order 3 that has one fixed point. Up to isomorphism there are 8,536 such symmetric (61,16,4) designs in the codes of D0;:::;D5, that together with their duals form a set of 17,058 mutually nonisomorphic designs. Further, from the code of one of the previously known designs, D2, we construct 148 designs invariant under an automorphism group of order 3 that has 16 fixed points, that together with their duals give 296 nonisomorphic designs. Hence, the obtained designs and their duals give us 17,354 symmetric (61,16,4) designs, of which 17,350 are new. Information about the full automorphism groups of these designs is given in Table 1. Since the designs D1 and D4 are not among the newly constructed designs, these results improve the lower bound on the number of nonisomorphic symmetric (61,16,4) designs from 6 to 17,356. A list of 17,354 symmetric (61,16,4) designs constructed from the codes of the designs D0,....,D5 is available at http://www.math.uniri.hr/~sanjar/structures/ New Symmetric (61,16,4) Designs Obtained from Codes 69

Table 1 Designs constructed Aut.D/j Structure of Aut.D/ No. designs from the codes C0;:::;C5 270 . / 1 and their dual designs E9 W Z3 D10 108 .E9 W Z3/ E4 3 54 .E9 W Z3/ Z2 27 36 Z6 Z6 4 30 D10 Z3 2 27 E9 W Z3 35 18 Z6 Z3 84 12 Z6 Z2 280 9 E9 524 6 Z6 2;170 3 Z3 14;224

Acknowledgements The authors wish to thank the unknown referee for making several very useful suggestions. This work has been supported in part by Croatian Science Foundation under the project 1637. Vladimir Tonchev acknowledges support by an NSA research grant H98230-15- 1-0042.

References

1. Assmus, Jr., E.F., Key, J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992) [Cambridge Tracts in Mathematics, vol. 103 (Second printing with corrections, 1993)] 2. Beker, H.J., Mitchell, C.J.: A construction method for point divisible designs. J. Stat. Plann. Inference 2, 293–306 (1978) 3. Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999) 4. Bosma, W., Cannon, J.: Handbook of Magma Functions. Department of Mathematics, Univer- sity of Sydney (1994). http://magma.maths.usyd.edu.au/magma 5. Cepuli´ c,´ V.: The unique symmetric block design (61,16,4) admitting an automorphism of order 15 operating standardly. Discret. Math. 175, 259–263 (1997) 6. Cepuli´ c,´ V.: Symmetric block designs .61; 16; 4/ admitting an automorphism of order 15. Glas. Mat. Ser. III 35(55), 233–244 (2000) 7. Colbourn, C.J., Dinitz, J.F. (eds.): Handbook of Combinatorial Designs, 2nd edn. Chapman & Hall/CRC Press, Boca Raton (2007) 8. Hall, Jr., M.: Combinatorial Theory, 2nd edn. Wiley, New York (1986) 9. Landjev, I., Topalova, S.: New symmetric (61,16,4) designs invariant under the dihedral group of order 10. Serdica Math. J. 24, 179–186 (1998) 10. Lander, E.: Symmetric Designs: An Algebraic Approach. Cambridge University Press, Cam- bridge (1983) 11. Mitchell, C.J.: An inﬁnite family of symmetric designs. Discret. Math. 26, 247–250 (1979) 12. Munemasa, A., Tonchev, V.D.: A new quasi-symmetric 2-(56,16,6) design obtained from codes. Discret. Math. 284, 231–234 (2004) 13. Rajkundlia, D.P.: Some techniques for constructing inﬁnite families of BIBD’s. Discret. Math. 44, 61–96 (1983) 14. van Trung, T.: Symmetric designs. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, pp. 82. CRC Press, Boca Raton (1996) D-Optimal Matrices of Orders 118, 138, 150, 154 and 174

Dragomir Ž. -Dokovic´ and Ilias S. Kotsireas

Dedicated to Hadi Kharaghani on the occasion of his 70th birthday

Abstract We construct supplementary difference sets (SDSs) with parameters .59I 28; 22I 21/, .69I 31; 27I 24/, .75I 36; 29I 28/, .77I 34; 31I 27/ and .87I 38; 36I 31/. These SDSs give D-optimal designs (DO-designs) of two-circulant type of orders 118,138,150,154 and 174. Until now, no DO-designs of orders 138,154 and 174 were known. While a DO-design (not of two-circulant type) of order 150 was constructed previously by Holzmann and Kharaghani, no such design of two-circulant type was known. The smallest undecided order for DO-designs is now 198. We use a novel property of the compression map to speed up some computations.

Keywords D-optimal designs • Supplementary difference sets • Periodic autocorrelation function • Compression method

1 Introduction

Let v be any positive integer. We say that a sequence A D Œa0; a1;:::;av1 is a binary sequence if ai 2f1; 1g for all i. We denote by Zv Df0;1;:::;v 1g the ring of integers modulo v. There is a bijection from the set of all binary sequences of

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. D.Ž. -Dokovic´ Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1 e-mail: [email protected] I.S. Kotsireas () Department of Physics & Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5 e-mail: [email protected]

© Springer International Publishing Switzerland 2015 71 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_6 72 D.Ž. -Dokovic´ and I.S. Kotsireas length v to the set of all subsets of Zv which assigns to the sequence A the subset fi 2 Zv W ai D1g.IfX Â Zv, then the corresponding binary sequence Œx0; x1;:::;xv1 has xi D1 if i 2 X and xi DC1 otherwise. We associate with X the cyclic matrix CX of order v having this sequence as its ﬁrst row. Assume temporarily that v is odd. If A is a fC1; 1g-matrix of size 2v 2v,then it is well known [5, 9]that

det A Ä 2v.2v 1/.v 1/v1: (1)

Moreover, this inequality is strict if 2v 1 is not a sum of two squares [5,Satz5.4]. In this paper we are interested only in the case when equality holds in (1). In that case we say that A is a D-optimal design (DO-design) of order 2v. Hence, in the context of DO-designs we shall assume that 2v 1 is a sum of two squares. For the problem of maximizing the determinant of square fC1; 1g-matrices of any fixed order, we refer the interested reader to [1] and its references. Many DO-designs of order 2v can be constructed by using supplementary difference sets (SDSs) with suitable parameters .vI r; sI /. We recall that these parameters are nonnegative integers such that .v 1/ D r.r 1/ C s.s 1/. (See Sect. 2 below for the formal definition of SDSs over a finite cyclic group.) For convenience, we also introduce the parameter n D r C s . Without any loss of generality we may assume that the parameter set is normalized which means that we have v=2 r s 0. The SDSs that we need are those for which v D 2n C 1. We refer to them as D-optimal SDSs. The feasible parameter sets for the D-optimal SDSs can be easily generated by using the following proposition. Apparently this fact has not been observed so far. Proposition 1.1. Let P be the set of ordered pairs .x; y/ of integers x; y such that x y 0. Let Q be the set of normalized feasible parameter sets .vI r; sI / for D-optimal SDSs. Thus, it is required that v D 2n C 1 where n D r C s .Then the map P ! Q, given by the formulas

v D 1 C x.x C 1/ C y.y C 1/; (2) ! ! x C 1 y r ; D 2 C 2 (3) ! ! x y C 1 s ; D 2 C 2 (4) ! ! x y ; D 2 C 2 (5) is a bijection. xC1 yC1 We leave the proof to the interested reader. Note that n D 2 C 2 . D-Optimal Matrices of Orders 118, 138, 150, 154 and 174 73

Let .X; Y/ be a D-optimal SDS with parameters .vI r; sI /. Then the associated matrices CX and CY satisfy the equation T T .2v 2/ 2 ; CXCX C CY CY D Iv C Jv (6) where Iv is the identity matrix, Jv is the matrix with all entries equal to 1,andthe superscript T denotes the transposition of matrices. One can verify that the matrix Â Ã CX CY T T (7) CY CX is a DO-design of order 2v. We say that the DO-designs obtained by this construction (due to Ehlich and Wojtas) are of two-circulant type (2c type). In the range 0 < v < 100, v odd integer, the condition that 2v 1 is a sum of two squares rules out the following 14 odd integers: 11, 17, 29, 35, 39, 47, 53, 65, 67, 71, 81, 83, 89, 85. In the remaining 36 cases the DO-designs of order 2v are known (see [3, 8] and their references) except for v D 69; 77; 87; 99. In the case v D 75 the only known DO-design of order 150 [7] is not of 2c type. Our main result is the construction of DO-designs of 2c type for orders 2v with v D 59; 69; 75; 77; 87. This is accomplished by constructing SDSs with parameters .59I 28; 22I 21/, .69I 31; 27I 24/, .75I 36; 29I 28/, .77I 34; 31I 27/ and .87I 38; 36I 31/, respectively. These SDSs give DO-designs of 2c type of orders 118,138,150,154 and 174. Until now, no DO-designs of orders 138,154 and 174 were known, and no DO-design of 2c type and order 150 was known. However, a DO-design (not of 2c type) of order 150 was constructed previously by Holzmann and Kharaghani [7]. The first DO-design of order 118 was constructed in [6], we provide two more non-equivalent examples. The main tool that we use in our constructions is the method of compression of SDSs which we developed in our recent paper [4]. This method uses a nontrivial factorization v D md and so it can be applied only when v is a composite integer. In the cases mentioned above we used the factorizations with m D 3 or m D 7. In Sect. 2 we recall the definition of SDS over finite cyclic groups, and in Sect. 3 we establish a relationship between power density functions of a complex sequence of length v D md and its compressed sequence of length d. This relationship was used to speed up some of the computations. In Sect. 4 we list the 2, 19, 3, 1 and 3 nonequivalent SDSs for the DO-designs of order 118, 138, 150, 154 and 174, respectively. Consequently, for orders less than 200 only the DO-design of order 198 remains unknown. In some cases, for a given odd integer v such that 2v 1 is a sum of two squares, there exist more than one feasible parameter set .vI r; sI / with v D 2nC1 and v=2 r s (see [3, Table I]). For instance, this is the case for v D 85. In that case there are two feasible parameter sets and an SDS is known only for one of them. In the appendix we list D-optimal SDSs, one per the parameter set .vI r; sI /,for all v < 100 with two exceptions where such SDS is not known. Finally, we point out two misprints in our recent paper [4]. (i) The first formula in [4, eq. (16)] should readP ˇ0 D v.tv 4n/ C 4n. (ii) In item 4 of [4, Remark 1] 2 the formula should read .v 2ki/ D 4v. 74 D.Ž. -Dokovic´ and I.S. Kotsireas

2 Supplementary Difference Sets

We recall the deﬁnition of SDSs. Let k1;:::;kt be positive integers and an integer such that

Xt .v 1/ D ki.ki 1/: (8) iD1

Definition 2.1. We say that the subsets X1;:::;Xt of Zv with jXijDki for i 2 f1;:::;tg are supplementary difference sets (SDS) with parameters .vI k1;:::;ktI /, if for every nonzero element c 2 Zv there are exactly ordered triples .a; b; i/ such that fa; bgÂXi and a b D c .mod v/. These SDSs are defined over the cyclic group of order v, namely the additive group of the ring Zv. More generally SDSs can be defined over any finite abelian group, and there are also further generalizations where the group may be any finite group. However, in this paper we shall consider only the cyclic case. In the context of an SDS, say X1;:::;Xt, with parameters .vI k1;:::;ktI /,we refer to the subsets Xi as the base blocks and we introduce an additional parameter, n, defined by:

n D k1 CCkt : (9)

If x is an indeterminate, then the quotient ring CŒx=.xv 1/ is isomorphic to the ring of complex circulant matrices of order v. Under this isomorphism x corresponds to the cyclic matrix with ﬁrst row Œ0;1;0;0;:::;0. By applying this isomorphism to the identity [4, (13)], we obtain that the following matrix identity holds

Xt T 4 . v 4 / ; CiCi D nIv C t n Jv (10) iD1

where Ci D CXi is the cyclic matrix associated with Xi. In this paper we are mainly interested in SDSs .X; Y/ with two base blocks, i.e., t D 2.Thenifv D 2n C 1 the identity (10) reduces to the identity (6).

3 Compression of SDSs

Let A be a complex sequence of length v. For the standard definitions of periodic autocorrelation functions .PAF A/, discrete Fourier transform .DFTA/,power spectral density .PSDA/ of A, and the definition of complex complementary D-Optimal Matrices of Orders 118, 138, 150, 154 and 174 75 sequences, we refer the reader to our paper [4]. If we have a collection of complex complementary sequences of length v D dm, then we can compress them to obtain complementary sequences of length d. We refer to the ratio v=d D m as the compression factor. Here is the precise definition.

Deﬁnition 3.1. Let A D Œa0; a1;:::;av1 be a complex sequence of length v D dm and set

.d/ ::: ; 0;:::; 1: aj D aj C ajCd C C ajC.m1/d j D d (11)

.d/ Œ .d/; .d/;:::; .d/ Then we say that the sequence A D a0 a1 ad1 is the m-compression of A. Let X; Y be a D-optimal SDS with parameters .vI r; sI / and let n D r C s . Thus v D 2n C 1. Assume that v D md is a nontrivial factorization. Let A; B be their associated binary sequences. Then the m-compressed sequences A.d/; B.d/ form a complementary pair. In general they are not binary sequences, their terms belong to the set fm; m2;:::;mC2;mg. The search for such pairs X; Y is broken into two stages: first we construct the candidate complementary sequences A.d/; B.d/ of length d, and second we lift each of them and search to find the D-optimal pairs .X; Y/. Each of the stages requires a lot of computational resources. There are additional theoretical results that can be used to speed up these computations. Some of them are described in [4], namely we use “bracelets” and “charm bracelets” to speed up the first stage. We give below a new theoretical result, which we used to speed up the second stage.

Theorem 3.1. Let A D Œa0; a1;:::;av1 be a complex sequence of length v D md ; >1 .d/ Œ .d/; .d/;:::; .d/ where m d are integers. Let A D a0 a1 ad1 be the m-compression of the sequence A. Then

. / . /; 0;1;:::; 1: PSDA ms D PSDA.d/ s s D d (12)

Proof. For the discrete Fourier transform of A we have

Xv1 Xv1 Xd1 . / !mjs !js .d/!js; DFTA ms D aj D aj 0 D aj 0 (13) jD0 jD0 jD0

m where ! D exp.2i=v/ and !0 D ! D exp.2i=d/. Hence, by using the Wiener– Khinchin theorem (i.e., that PSD D DFT ı PAF), we have

2 PSDA.ms/ D jDFTA.ms/j Xd1 .d/!js .d/!ks D aj 0 ak 0 j;kD0 76 D.Ž. -Dokovic´ and I.S. Kotsireas

Xd1 .d/ .d/!.jk/s D aj ak 0 j;kD0 ! Xd1 Xd1 .d/ .d/ !rs D akCrak 0 rD0 kD0 Xd1 . /!rs D PAF A.d/ r 0 rD0 . /. / D DFT PAF A.d/ s . /: D PSDA.d/ s

4 Computational Results for DO-Designs

All solutions are in the canonical form deﬁned in [2] and since they are different, this implies that they are pairwise nonequivalent. Taking into account our new results, the open cases for DO-designs with v < 200 are

99; 111; 115; 117; 123; 129; 135; 139; 141; 147; 153; 159; 163; 167; 169; 175; 177; 185; 187; 189; 195; 199:

4.1 D-Optimal SDSs with Parameters .59I 28; 22I 21/

An SDSs with these parameters has been constructed in [6], it is equivalent to the one listed in the appendix. We have constructed two more such SDS, not equivalent to the one mentioned above.

.1/ f0; 1; 2; 4; 5; 6; 8; 9; 10; 12; 14; 19; 21; 24; 25; 28; 30; 31; 33; 37; 41; 42; 43; 45; 46; 52; 53; 54g; f0; 1; 2; 3; 5; 6; 7; 8; 13; 15; 16; 18; 21; 23; 27; 31; 32; 35; 38; 41; 48; 52g; .2/ f0; 1; 2; 3; 5; 7; 8; 11; 13; 14; 15; 17; 18; 19; 23; 25; 26; 31; 32; 33; 35; 38; 40; 42; 47; 51; 53; 56g; f0; 1; 3; 4; 5; 6; 8; 13; 14; 15; 17; 23; 25; 26; 29; 30; 33; 36; 40; 41; 45; 46g: D-Optimal Matrices of Orders 118, 138, 150, 154 and 174 77

4.2 D-Optimal SDSs with Parameters .69I 31; 27I 24/

Until now, 69 was the smallest positive odd integer v for which the existence of DO- designs of order 2v was undecided. We have constructed 19 nonequivalent SDSs for the above parameter set, which give 19 DO-designs of order 138.

.1/ f0; 1; 3; 4; 6; 9; 10; 11; 13; 14; 17; 18; 20; 22; 26; 28; 29; 32; 33; 34; 39; 41; 43; 45; 46; 48; 51; 59; 60; 62; 63g; f0; 2; 3; 4; 8; 9; 10; 11; 12; 15; 16; 17; 21; 25; 26; 32; 33; 35; 36; 37; 39; 41; 46; 51; 54; 57; 59g; .2/ f0; 1; 3; 4; 5; 6; 7; 8; 9; 10; 12; 16; 18; 19; 23; 25; 27; 28; 31; 32; 33; 39; 40; 41; 42; 47; 52; 53; 58; 60; 63g; f0; 1; 3; 4; 5; 8; 10; 12; 13; 14; 18; 20; 23; 26; 27; 30; 31; 38; 41; 43; 44; 47; 51; 53; 55; 58; 63g; .3/ f0; 2; 3; 5; 6; 7; 8; 11; 12; 14; 15; 16; 21; 22; 23; 24; 26; 30; 31; 32; 35; 36; 37; 41; 43; 46; 49; 53; 54; 56; 58g; f0; 1; 2; 3; 4; 8; 9; 11; 14; 16; 17; 20; 24; 28; 31; 33; 35; 37; 38; 41; 42; 45; 47; 53; 57; 59; 60g; .4/ f0; 1; 2; 4; 5; 6; 7; 8; 10; 12; 15; 17; 20; 23; 24; 25; 28; 30; 34; 36; 37; 40; 42; 46; 47; 48; 49; 51; 55; 62; 63g; f0; 1; 2; 3; 4; 5; 7; 11; 12; 14; 16; 18; 19; 21; 22; 28; 31; 32; 37; 38; 43; 47; 51; 52; 55; 60; 63g; .5/ f0; 1; 2; 3; 4; 7; 8; 10; 11; 13; 16; 18; 21; 24; 25; 26; 27; 30; 32; 33; 34; 37; 39; 41; 44; 45; 54; 55; 58; 59; 60g; f0; 1; 2; 5; 6; 8; 10; 11; 12; 14; 15; 17; 23; 24; 30; 32; 34; 36; 39; 40; 43; 44; 51; 56; 59; 61; 63g; .6/ f0; 1; 3; 4; 5; 6; 7; 10; 14; 15; 16; 18; 22; 24; 25; 26; 27; 28; 32; 33; 34; 39; 41; 44; 48; 52; 53; 55; 57; 60; 61g; f0; 2; 3; 5; 6; 8; 11; 12; 13; 15; 18; 19; 23; 25; 26; 28; 33; 37; 39; 40; 42; 43; 44; 48; 52; 58; 64g; .7/ f0; 1; 2; 3; 4; 5; 7; 10; 11; 13; 14; 19; 20; 22; 23; 25; 28; 30; 31; 33; 35; 36; 37; 41; 43; 45; 50; 54; 57; 58; 64g; f0; 1; 2; 3; 4; 6; 7; 11; 12; 15; 20; 22; 25; 26; 27; 30; 31; 32; 38; 39; 42; 44; 46; 48; 55; 59; 62g; .8/ f0; 2; 3; 5; 6; 7; 9; 10; 11; 12; 14; 15; 18; 20; 22; 24; 25; 29; 34; 35; 36; 37; 38; 45; 46; 49; 51; 53; 55; 59; 66g; f0; 1; 2; 3; 5; 6; 11; 13; 14; 17; 20; 21; 22; 27; 28; 29; 33; 34; 38; 41; 43; 46; 50; 52; 53; 56; 64g; .9/ f0; 1; 2; 3; 4; 5; 6; 8; 9; 10; 13; 14; 15; 16; 20; 22; 25; 27; 28; 32; 35; 36; 37; 43; 45; 46; 49; 52; 54; 56; 61g; f0; 1; 3; 5; 7; 8; 11; 13; 14; 15; 19; 23; 26; 28; 29; 30; 33; 39; 43; 44; 45; 49; 52; 57; 60; 61; 63g; .10/ f0; 1; 2; 3; 4; 6; 7; 8; 13; 14; 17; 19; 21; 22; 25; 26; 28; 29; 30; 34; 37; 40; 41; 42; 44; 45; 50; 51; 54; 59; 64g; f0; 1; 3; 4; 5; 6; 7; 10; 13; 15; 16; 17; 22; 24; 26; 31; 33; 34; 37; 39; 40; 45; 47; 55; 57; 59; 65g; .11/ f0; 1; 2; 4; 5; 6; 7; 8; 11; 14; 15; 18; 19; 20; 21; 23; 25; 28; 29; 30; 37; 39; 41; 42; 43; 45; 47; 50; 54; 57; 62g; f0; 1; 2; 3; 5; 8; 9; 11; 13; 16; 17; 21; 24; 26; 27; 30; 33; 36; 40; 41; 42; 47; 51; 52; 53; 62; 64g; .12/ f0; 1; 2; 3; 4; 6; 8; 9; 10; 11; 13; 15; 16; 17; 20; 23; 24; 28; 29; 31; 34; 38; 39; 40; 43; 49; 51; 53; 55; 56; 59g; f0; 1; 2; 4; 6; 9; 10; 12; 13; 14; 18; 19; 23; 26; 30; 33; 35; 36; 44; 45; 47; 50; 51; 52; 58; 60; 63g; .13/ f0; 1; 2; 3; 4; 6; 7; 9; 11; 13; 16; 17; 20; 25; 26; 27; 28; 30; 32; 35; 36; 37; 40; 43; 46; 47; 52; 57; 58; 60; 64g; f0; 1; 3; 4; 5; 9; 10; 12; 13; 17; 18; 19; 20; 21; 25; 27; 31; 32; 34; 38; 46; 48; 50; 51; 54; 56; 59g; .14/ f0; 1; 2; 3; 4; 5; 6; 8; 10; 13; 14; 15; 16; 18; 20; 23; 24; 26; 29; 32; 33; 36; 39; 41; 43; 48; 50; 53; 54; 55; 61g; f0; 1; 2; 3; 6; 7; 8; 9; 13; 15; 17; 22; 23; 26; 30; 33; 34; 37; 42; 45; 46; 48; 50; 51; 59; 60; 65g; .15/ f0; 1; 2; 3; 5; 6; 7; 8; 11; 15; 18; 20; 21; 23; 25; 29; 30; 31; 32; 38; 39; 41; 42; 43; 44; 49; 51; 55; 57; 60; 65g; f0; 1; 2; 4; 5; 8; 9; 11; 12; 13; 17; 20; 23; 24; 26; 28; 30; 33; 34; 37; 39; 40; 47; 48; 53; 55; 65g; .16/ f0; 1; 2; 4; 5; 8; 10; 11; 14; 15; 16; 18; 19; 22; 23; 25; 28; 29; 30; 34; 37; 38; 40; 42; 45; 47; 50; 52; 53; 54; 63g; f0; 1; 2; 3; 5; 6; 9; 14; 16; 17; 18; 19; 22; 26; 28; 30; 32; 37; 38; 39; 44; 45; 47; 49; 50; 56; 65g; .17/ f0; 1; 2; 3; 4; 6; 7; 8; 12; 14; 15; 17; 22; 23; 24; 26; 27; 28; 30; 33; 37; 40; 41; 45; 48; 51; 54; 56; 57; 58; 64g; f0; 1; 2; 5; 6; 7; 11; 13; 14; 15; 17; 21; 23; 26; 30; 31; 33; 35; 37; 38; 40; 42; 43; 48; 51; 52; 60g; 78 D.Ž. -Dokovic´ and I.S. Kotsireas

.18/ f0; 1; 2; 3; 6; 7; 8; 9; 12; 13; 15; 20; 21; 23; 24; 26; 28; 30; 31; 32; 33; 35; 38; 42; 43; 44; 48; 52; 56; 59; 62g; f0; 1; 2; 4; 5; 6; 8; 9; 11; 17; 18; 19; 21; 23; 28; 32; 33; 37; 40; 43; 44; 46; 48; 53; 54; 56; 62g; .19/ f0; 1; 2; 3; 7; 8; 9; 10; 11; 12; 13; 16; 18; 19; 22; 24; 25; 27; 29; 32; 35; 36; 39; 41; 45; 46; 50; 52; 54; 56; 57g; f0; 1; 2; 3; 4; 6; 8; 12; 15; 16; 20; 21; 22; 24; 29; 30; 34; 35; 38; 41; 42; 45; 48; 50; 53; 58; 60g:

4.3 D-Optimal SDSs with Parameters .75I 36; 29I 28/

Until now, no DO-design of 2c type and order 150 was known. We have constructed three nonequivalent SDSs for the above parameter set. They give three DO-designs of 2c type and order 150.

.1/ f0; 1; 2; 3; 4; 5; 8; 9; 10; 12; 13; 16; 17; 19; 22; 25; 27; 28; 30; 32; 33; 34; 38; 40; 42; 44; 47; 49; 51; 54; 57; 60; 61; 65; 66; 67g; f0; 1; 2; 4; 5; 6; 7; 9; 10; 12; 16; 17; 21; 24; 25; 30; 31; 32; 35; 38; 39; 41; 43; 45; 51; 52; 61; 63; 64g; .2/ f0; 1; 2; 3; 7; 8; 9; 10; 11; 13; 15; 16; 20; 21; 23; 24; 26; 27; 30; 32; 34; 36; 38; 39; 44; 45; 48; 49; 50; 52; 55; 60; 64; 65; 66; 69g; f0; 2; 3; 4; 5; 6; 7; 9; 12; 13; 14; 17; 21; 22; 24; 26; 31; 34; 37; 39; 40; 46; 50; 53; 54; 55; 57; 61; 69g; .3/ f0; 1; 4; 5; 6; 7; 9; 12; 14; 15; 16; 17; 20; 22; 24; 25; 26; 28; 30; 32; 33; 39; 40; 41; 44; 45; 46; 47; 49; 53; 56; 59; 62; 65; 67; 69g; f0; 2; 4; 5; 6; 7; 9; 11; 12; 16; 18; 19; 22; 23; 29; 30; 31; 33; 34; 40; 43; 44; 48; 49; 52; 53; 58; 60; 61g:

4.4 D-Optimal SDSs with Parameters .77I 34; 31I 27/

We have constructed only one solution.

.1/ f0; 2; 3; 4; 5; 6; 9; 10; 12; 14; 17; 19; 22; 23; 24; 26; 29; 30; 32; 33; 36; 37; 39; 44; 45; 48; 50; 54; 58; 60; 61; 63; 69; 71g; f0; 1; 2; 4; 5; 6; 9; 10; 12; 14; 17; 20; 21; 22; 23; 24; 28; 29; 35; 38; 40; 44; 45; 49; 51; 52; 53; 54; 60; 64; 65g: D-Optimal Matrices of Orders 118, 138, 150, 154 and 174 79

4.5 D-Optimal SDSs with Parameters .87I 38; 36I 31/

We have constructed three nonequivalent solutions.

.1/ f0; 1; 2; 3; 4; 5; 6; 8; 10; 12; 16; 18; 22; 23; 24; 25; 32; 33; 36; 37; 38; 39; 43; 46; 47; 50; 54; 56; 57; 61; 62; 63; 66; 69; 71; 74; 80; 83g; f0; 1; 2; 5; 6; 8; 10; 11; 13; 17; 18; 19; 21; 23; 24; 26; 27; 29; 33; 36; 38; 40; 43; 45; 48; 49; 51; 52; 53; 54; 58; 65; 66; 69; 77; 78g; .2/ f0; 1; 2; 4; 5; 7; 10; 11; 14; 15; 17; 19; 22; 23; 24; 25; 27; 29; 30; 35; 36; 39; 42; 44; 50; 51; 54; 55; 57; 59; 61; 65; 66; 68; 73; 77; 78; 81g; f0; 1; 2; 3; 4; 5; 6; 7; 8; 13; 14; 19; 21; 22; 27; 28; 30; 31; 32; 36; 38; 39; 40; 43; 45; 47; 48; 49; 54; 57; 59; 61; 67; 70; 73; 77g; .3/ f0; 1; 3; 5; 6; 8; 9; 11; 12; 15; 16; 18; 19; 20; 25; 27; 28; 29; 31; 33; 40; 41; 45; 46; 47; 50; 51; 55; 58; 61; 62; 64; 68; 69; 70; 72; 76; 78g; f0; 1; 2; 3; 4; 7; 8; 10; 12; 14; 15; 17; 19; 24; 27; 28; 29; 33; 34; 35; 36; 37; 40; 42; 47; 48; 50; 51; 53; 58; 63; 66; 67; 69; 78; 82g:

Acknowledgements The authors wish to acknowledge generous support by NSERC. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET) and Compute/Calcul Canada. We thank a referee for his suggestions.

Appendix: D-Optimal SDSs with v < 100

We list here all D-optimal parameter sets .vI r; sI / with v=2 r s and v < 100 and for each of them (with two exceptions) we give one DO-design of 2c type by recording the two base blocks of the corresponding SDS. In the two exceptional cases we indicate by a question mark that such designs are not yet known. In particular, this means that DO-designs of order 2v < 200, with v odd, exist for all feasible orders (those for which 2v 1 is a sum of two squares) except for v D 99. This list will be useful to interested readers as examples of such designs are spread out over many papers in the literature. For the beneﬁt of the readers interested in binary sequences we mention that these SDSs give two binary sequences of length v with PAF +2, i.e., D-optimal matrices. 80 D.Ž. -Dokovic´ and I.S. Kotsireas

.vI r; sI / Base blocks .3I 1; 0I 0/ f0g; ; .5I 1; 1I 0/ f0g; f0g .7I 3; 1I 1/ f0; 1; 3g; f0g .9I 3; 2I 1/ f0; 1; 4g; f0; 2g .13I 4;4I 2/ f0; 1; 4; 6g; f0; 1; 4; 6g .13I 6; 3I 3/ f0; 1; 2; 4; 7; 9g; f0; 1; 4g .15I 6; 4I 3/ f0; 1; 2; 4; 6; 9g; f0; 1; 4; 9g .19I 7; 6I 4/ f0; 1; 2; 3; 7; 11; 14g; f0; 2; 5; 6; 9; 11g .21I 10; 6I 6/ f0; 1; 2; 3; 4; 6; 8; 11; 12; 16g; f0; 1; 3; 7; 10; 15g .23I 10; 7I 6/ f0; 1; 3; 4; 5; 7; 8; 12; 14; 18g; f0; 1; 2; 7; 9; 12; 15g .25I 9; 9I 6/ f0; 1; 2; 4; 7; 11; 14; 15; 20g; f0; 1; 2; 4; 6; 9; 10; 12; 17g .27I 11; 9I 7/ f0; 1; 3; 4; 5; 9; 10; 11; 13; 16; 19g; f0; 1; 2; 4; 8; 12; 15; 17; 22g .31I 15; 10I 10/ f0; 1; 2; 3; 5; 6; 7; 11; 13; 15; 16; 18; 23; 24; 27g; f0; 2; 3; 5; 6; 8; 12; 19; 20; 27g .33I 13; 12I 9/ f0; 1; 2; 4; 5; 6; 8; 10; 15; 17; 20; 25; 26g; f0; 2; 3; 5; 6; 9; 12; 13; 17; 19; 24; 25g .33I 15; 11I 10/ f0; 1; 2; 3; 4; 5; 8; 10; 12; 13; 14; 18; 19; 22; 26g; f0; 1; 2; 5; 8; 11; 15; 17; 20; 22; 28g .37I 16; 13I 11// f0; 1; 2; 3; 4; 7; 8; 11; 13; 15; 16; 18; 23; 24; 27; 33g; f0; 1; 2; 4; 8; 10; 13; 14; 18; 20; 21; 23; 32g .41I 16; 16I 12/ f0; 1; 2; 3; 5; 7; 8; 9; 13; 18; 19; 22; 23; 26; 32; 34g; f0; 1; 3; 4; 6; 8; 11; 13; 15; 16; 17; 23; 24; 27; 30; 36g .43I 18; 16I 13/ f0; 1; 2; 3; 4; 7; 9; 11; 12; 13; 16; 19; 22; 24; 25; 29; 30; 36g; f0; 1; 2; 4; 5; 6; 9; 14; 16; 17; 20; 24; 26; 31; 33; 39g .43I 21; 15I 15/ f0; 1; 2; 3; 4; 5; 6; 7; 11; 12; 13; 14; 17; 20; 24; 25; 28; 30; 31; 34; 39g; f0; 2; 3; 4; 7; 9; 12; 14; 16; 22; 24; 30; 31; 34; 39g .45I 21; 16I 15/ f0; 1; 2; 3; 5; 6; 8; 10; 12; 13; 14; 20; 21; 22; 25; 28; 29; 32; 34; 35; 42g; f0; 1; 2; 4; 5; 6; 10; 11; 14; 16; 19; 22; 29; 31; 33; 40g .49I 22; 18I 16/ f0; 1; 2; 3; 4; 5; 6; 9; 11; 13; 14; 19; 20; 21; 23; 26; 27; 30; 35; 38; 40; 42g; f0; 1; 3; 4; 5; 8; 9; 13; 15; 19; 21; 24; 26; 27; 30; 37; 43; 44g .51I 21; 20I 16/ f0; 2; 4; 5; 6; 9; 11; 12; 13; 18; 19; 21; 22; 26; 27; 28; 30; 33; 38; 39; 41g; f0; 1; 2; 4; 5; 6; 9; 10; 12; 14; 17; 22; 24; 25; 28; 31; 35; 37; 41; 42g .55I 24; 21I 18/ f0; 1; 2; 3; 6; 8; 10; 11; 13; 14; 17; 19; 20; 21; 24; 26; 28; 29; 33; 34; 40; 41; 43; 44g; f0; 1; 2; 3; 6; 7; 9; 11; 12; 15; 19; 21; 25; 29; 34; 36; 37; 38; 40; 45; 50g .57I 28; 21I 21/ f0; 1; 2; 3; 4; 5; 8; 9; 10; 11; 13; 16; 17; 19; 21; 22; 23; 24; 27; 31; 34; 36; 37; 38; 41; 43; 49; 50g; f0; 1; 3; 4; 7; 9; 11; 13; 15; 16; 20; 25; 26; 29; 30; 35; 37; 40; 41; 43; 48g D-Optimal Matrices of Orders 118, 138, 150, 154 and 174 81

.vI r; sI / Base blocks .59I 28; 22I 21/ f0; 2; 3; 5; 6; 8; 9; 10; 13; 15; 16; 17; 19; 23; 25; 26; 27; 29; 30; 34; 38; 39; 41; 43; 44; 45; 53; 56g; f0; 1; 2; 3; 5; 7; 8; 10; 12; 13; 19; 20; 22; 24; 28; 32; 33; 37; 38; 44; 45; 51g .61I 25; 25I 20/ f0; 2; 4; 7; 8; 9; 10; 12; 13; 18; 20; 23; 24; 25; 26; 29; 32; 33; 34; 38; 41; 44; 48; 51; 52g; f0; 1; 2; 4; 6; 7; 8; 12; 13; 14; 15; 16; 19; 23; 29; 30; 32; 34; 36; 39; 41; 44; 49; 50; 53g .63I 27; 25I 21/ f0; 1; 2; 3; 5; 7; 10; 11; 12; 15; 18; 21; 23; 24; 25; 26; 31; 32; 36; 37; 40; 43; 44; 47; 49; 51; 53g; f0; 2; 4; 6; 7; 8; 9; 10; 11; 12; 16; 20; 21; 24; 27; 30; 33; 38; 39; 40; 45; 47; 55; 56; 60g .63I 29; 24I 22/ f0; 1; 2; 3; 4; 6; 7; 11; 12; 13; 14; 20; 21; 22; 25; 26; 27; 30; 33; 35; 36; 38; 39; 42; 46; 48; 50; 53; 57g; f0; 1; 3; 5; 7; 8; 10; 11; 13; 14; 16; 18; 19; 23; 30; 33; 34; 35; 39; 40; 48; 52; 54; 56g .69I 31; 27I 24/ f0; 1; 3; 4; 6; 9; 10; 11; 13; 14; 17; 18; 20; 22; 26; 28; 29; 32; 33; 34; 39; 41; 43; 45; 46; 48; 51; 59; 60; 62; 63g; f0; 2; 3; 4; 8; 9; 10; 11; 12; 15; 16; 17; 21; 25; 26; 32; 33; 35; 36; 37; 39; 41; 46; 51; 54; 57; 59g .73I 31; 30I 25/ f0; 1; 2; 3; 4; 5; 7; 9; 11; 12; 16; 17; 21; 22; 25; 26; 30; 32; 34; 37; 38; 43; 44; 45; 46; 49; 52; 54; 56; 59; 62g; f0; 1; 3; 4; 7; 8; 9; 11; 15; 16; 17; 18; 21; 23; 26; 27; 28; 29; 31; 33; 40; 42; 46; 47; 50; 53; 56; 62; 63; 65g .73I 36; 28I 28/ f0; 1; 3; 4; 6; 7; 9; 10; 12; 13; 14; 15; 19; 20; 21; 25; 27; 28; 29; 30; 31; 36; 38; 39; 41; 42; 43; 46; 50; 51; 54; 55; 57; 59; 61; 63g; f0; 1; 4; 6; 7; 11; 13; 14; 18; 20; 21; 22; 23; 24; 26; 30; 31; 35; 38; 40; 48; 51; 53; 54; 58; 59; 63; 65g .75I 36; 29I 28/ f0; 1; 2; 3; 4; 5; 8; 9; 10; 12; 13; 16; 17; 19; 22; 25; 27; 28; 30; 32; 33; 34; 38; 40; 42; 44; 47; 49; 51; 54; 57; 60; 61; 65; 66; 67g; f0; 1; 2; 4; 5; 6; 7; 9; 10; 12; 16; 17; 21; 24; 25; 30; 31; 32; 35; 38; 39; 41; 43; 45; 51; 52; 61; 63; 64g .77I 34; 31I 27/ f0; 2; 3; 4; 5; 6; 9; 10; 12; 14; 17; 19; 22; 23; 24; 26; 29; 30; 32; 33; 36; 37; 39; 44; 45; 48; 50; 54; 58; 60; 61; 63; 69; 71g; f0; 1; 2; 4; 5; 6; 9; 10; 12; 14; 17; 20; 21; 22; 23; 24; 28; 29; 35; 38; 40; 44; 45; 49; 51; 52; 53; 54; 60; 64; 65g .79I 37; 31I 29/ f0; 1; 2; 3; 4; 5; 6; 9; 12; 13; 14; 16; 18; 23; 24; 30; 31; 32; 33; 35; 38; 39; 40; 44; 46; 48; 52; 53; 56; 57; 58; 61; 64; 67; 69; 72; 73g; f0; 1; 3; 4; 6; 8; 10; 11; 13; 14; 15; 17; 21; 22; 27; 28; 30; 32; 33; 34; 37; 44; 46; 47; 50; 52; 53; 55; 65; 69; 75g .85I 36; 36I 30/ f0; 1; 2; 3; 5; 6; 8; 9; 12; 13; 15; 22; 24; 26; 28; 29; 33; 34; 35; 36; 38; 40; 41; 46; 48; 49; 51; 52; 56; 57; 60; 66; 70; 75; 78; 80g; f0; 2; 3; 4; 5; 6; 8; 11; 12; 17; 18; 19; 20; 21; 22; 25; 29; 31; 33; 36; 37; 38; 42; 43; 46; 47; 55; 57; 58; 61; 64; 66; 68; 73; 74; 81g .85I 39; 34I 31/ ‹ .87I 38; 36I 31/ f0; 1; 2; 3; 4; 5; 6; 8; 10; 12; 16; 18; 22; 23; 24; 25; 32; 33; 36; 37; 38; 39; 43; 46; 47; 50; 54; 56; 57; 61; 62; 63; 66; 69; 71; 74; 80; 83g; f0; 1; 2; 5; 6; 8; 10; 11; 13; 17; 18; 19; 21; 23; 24; 26; 27; 29; 33; 36; 38; 40; 43; 45; 48; 49; 51; 52; 53; 54; 58; 65; 66; 69; 77; 78g 82 D.Ž. -Dokovic´ and I.S. Kotsireas

.vI r; sI / Base blocks .91I 45; 36I 36/ f0; 2; 4; 5; 6; 8; 9; 10; 11; 12; 13; 14; 17; 18; 19; 21; 24; 25; 27; 30; 33; 34; 35; 36; 37; 38; 44; 45; 47; 48; 51; 52; 56; 57; 59; 64; 66; 67; 69; 71; 74; 75; 80; 84; 85g; f0; 2; 4; 6; 9; 10; 11; 14; 15; 16; 20; 22; 24; 27; 29; 31; 32; 34; 37; 38; 46; 49; 50; 51; 52; 53; 60; 63; 64; 66; 69; 70; 72; 76; 77; 85g .93I 42; 38I 34/ f0; 1; 4; 5; 6; 7; 8; 10; 15; 16; 17; 19; 22; 23; 26; 29; 30; 32; 33; 34; 35; 38; 40; 41; 45; 46; 47; 49; 53; 55; 60; 63; 65; 66; 70; 72; 73; 74; 77; 80; 82; 84g; f0; 1; 2; 3; 4; 6; 8; 10; 11; 12; 13; 15; 16; 22; 24; 26; 27; 30; 31; 32; 35; 40; 44; 47; 48; 49; 52; 53; 54; 60; 62; 64; 67; 70; 73; 82; 83; 88g .93I 45; 37I 36/ f0; 2; 3; 4; 6; 7; 8; 9; 11; 13; 14; 16; 18; 19; 20; 22; 23; 24; 26; 31; 34; 35; 37; 38; 39; 41; 43; 44; 47; 52; 53; 55; 59; 62; 63; 64; 66; 69; 70; 74; 75; 76; 81; 83; 86g; f0; 1; 2; 3; 6; 7; 10; 11; 12; 15; 16; 18; 19; 20; 26; 28; 29; 30; 33; 36; 40; 42; 51; 52; 53; 55; 57; 58; 60; 65; 66; 74; 77; 79; 80; 85; 87g .97I 46; 39I 37/ f0; 1; 2; 4; 6; 7; 8; 9; 11; 12; 14; 15; 17; 21; 22; 24; 25; 26; 28; 29; 34; 35; 36; 38; 44; 45; 47; 49; 51; 52; 53; 55; 57; 63; 64; 67; 68; 69; 73; 76; 78; 81; 82; 83; 86; 94g; f0; 1; 2; 3; 6; 8; 11; 12; 16; 17; 18; 20; 23; 25; 27; 28; 29; 30; 36; 37; 38; 41; 44; 45; 49; 51; 57; 60; 61; 62; 63; 64; 67; 69; 73; 76; 83; 91; 94g .99I 43; 42I 36/ ‹

References

1. Brent, R.P.: Finding many D-optimal designs by randomised decomposition and switching. Australas J. Comb. 55, 15–30 (2013) 2. -Dokovic,´ D.Ž.: Cyclic .vI r; sI / difference families with two base blocks and v Ä 50. Ann. Comb. 15, 233–254 (2011) 3. -Dokovic,´ D.Ž., Kotsireas, I.S.: New results on D-optimal matrices. J. Comb. Des. 20, 278–289 (2012) 4. -Dokovic,´ D.Ž., Kotsireas, I.S.: Compression of periodic complementary sequences and applications. Des. Codes Crypt. 74, 365–377 (2015) 5. Ehlich, H.: Determinantenabschätzungen für binäre Matrizen. Math. Z. 83, 123–132 (1964) 6. Fletcher, R.J., Koukouvinos, C., Seberry, J.: New skew-Hadamard matrices of order 4.59 and new D-optimal designs of order 2.59. Discret. Math. 286, 251–253 (2004) 7. Holzmann, W.H., Kharaghani, H.: A D-optimal design of order 150. Discret. Math 190, 265–269 (1998) 8. Kharaghani, H., Orrick, W.: D-optimal matrices. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, pp. 296–298, 2nd edn. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton (2007) 9. Wojtas, W.: On Hadamard’s inequality for the determinants of order non-divisible by 4. Colloq. Math. 12, 73–83 (1964) Periodic Golay Pairs of Length 72

Dragomir Ž. -Dokovic´ and Ilias S. Kotsireas

Dedicated to Hadi Kharaghani on the occasion of his 70th birthday

Abstract We construct supplementary difference sets (SDSs) with parameters .72I 36; 30I 30/. These SDSs give periodic Golay pairs of length 72. No periodic Golay pair of length 72 was known previously. The smallest undecided order for periodic Golay pairs is now 90. The periodic Golay pairs constructed here are the ﬁrst examples having length divisible by a prime congruent to 3 modulo 4. The main tool employed is a recently introduced compression method. We observe that Turyn’s multiplication of Golay pairs can be also used to multiply a Golay pair and a periodic Golay pair.

Keywords Ordinary and periodic Golay pairs • Supplementary difference sets • Compression method

1 Introduction

© Springer International Publishing Switzerland 2015 83 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_7 84 D.Ž. -Dokovic´ and I.S. Kotsireas length v to the set of all subsets of Zv which assigns to the sequence A the subset fi 2 Zv W ai D1g.IfX Â Zv, then the corresponding binary sequence Œx0; x1;:::;xv1 has xi D1 if i 2 X and xi DC1 otherwise. We associate with X the cyclic matrix CX of order v having this sequence as its first row. Periodic Golay pairs are periodic analogs of the well-known Golay pairs. Let us give a precise definition. For any complex sequence A D Œa0; a1;:::;av1, its periodic autocorrelation is a complex valued function PAFA W Zv ! C defined by

Xv1 PAF A.s/ D ajCsaNj; (1) jD0 where the indexes are computed modulo v and aN is the complex conjugate of a.A pair of binary sequences .A; B/ of length v is a periodic Golay pair if PAFA.j/ C PAF B.j/ D 0 for j ¤ 0. For more information on these pairs, see [6]. The length v of a periodic Golay pair must be even except for the trivial case v D 1, Many periodic Golay pairs of even length v can be constructed by using supplementary difference sets (SDSs) with suitable parameters .vI r; sI /. We recall that these parameters are nonnegative integers such that .v1/ D r.r1/Cs.s1/. (See Sect. 2 below for the formal deﬁnition of SDSs over a ﬁnite cyclic group.) For convenience, we also introduce the parameter n D r C s . Without any loss of generality we may assume that the parameter set is normalized which means that we have v=2 r s 0. The SDSs that we need are those for which v D 2n.We refer to them as periodic Golay SDSs. The feasible parameter sets for the periodic Golay SDSs can be easily generated by using the following proposition. Proposition 1.1. Let P be the set of ordered pairs .x; y/ of integers x; y such that x y 0 and x >0. Let Q be the set of normalized feasible parameter sets .vI r; sI /, with v even, for periodic Golay SDSs. Thus, it is required that v D 2n where n D r C s . Then the map P ! Q given by the formula

.x; y/ ! .2.x2 C y2/I x2 C y2 y; x2 C y2 xI x2 C y2 x y/ is a bijection. Proof. The inverse map Q ! P is given by v v .v r; s / . s; r/: I I ! 2 2

Note that n D x2 C y2. If .A; B/ is a periodic Golay pair of length v, then the corresponding pair of subsets .X; Y/ of Zv is an SDSs. In the nontrivial cases .v > 1/, the parameters .vI r; sI / satisfy the equation v D 2n. Recall that n D rCs. The converse is also true, i.e., if .X; Y/ is an SDSs with parameters .vI r; sI /, then the corresponding binary sequences .A; B/ form a periodic Golay pair of length v. Moreover, if a D v 2r and b D v 2s,thena2 C b2 D 2v. In particular, v must be even and a sum of two squares. The associated matrices CX and CY satisfy the equation Periodic Golay Pairs of Length 72 85

T T 2v : CXCX C CY CY D Iv (2)

Our main result is the construction of several periodic Golay pairs of length 72. This is accomplished by constructing the SDSs with parameters .72I 36; 30I 30/. The main tool that we use in the construction is the method of compression of SDSs developed in [4]. This method uses a nontrivial factorization v D md and so it can be applied only when v is a composite integer. In this case we used the factorization with m D 3 and d D 24. In Sect. 2 we recall the definition of SDSs over finite cyclic groups, and in Sect. 3 we establish a relationship between power spectral density functions of a complex sequence of length v D md and its compressed sequence of length d.This relationship was used to speed up some of the computations. In Sect. 5 we list eight nonequivalent SDSs which give eight periodic Golay pairs of length 72. This provides the first examples of periodic Golay pairs whose length is divisible by a prime congruent to 3 modulo 4.

2 Supplementary Difference Sets

We recall the deﬁnition of SDSs. Let k1;:::;kt be positive integers and an integer such that

Xt .v 1/ D ki.ki 1/: (3) iD1

Definition 2.1. We say that the subsets X1;:::;Xt of Zv with jXijDki for i 2f1;:::;tg are supplementary difference sets (SDSs) with parameters .vI k1;:::;ktI /, if for every nonzero element c 2 Zv there are exactly ordered triples .a; b; i/ such that fa; bgÂXi and a b D c .mod v/. These SDSs are defined over the cyclic group of order v, namely the additive group of the ring Zv. More generally SDSs can be defined over any finite abelian group, and there are also further generalizations where the group may be any finite group. However, in this paper we shall consider only the cyclic case. In the context of an SDSs, say X1;:::;Xt, with parameters .vI k1;:::;ktI /,we refer to the subsets Xi as the base blocks and we introduce an additional parameter, n,definedby

n D k1 CCkt : (4)

Xt T 4 . v 4 / ; CiCi D nIv C t n Jv (5) iD1 where Ci D CXi is the cyclic matrix associated with Xi. In this paper we are mainly interested in SDSs .X; Y/ with two base blocks .t D 2/ and v D 2n Then the identity (5) reduces to the identity (2).

3 Compression of SDSs

Let A be a complex sequence of length v. For the standard definitions of periodic autocorrelation functions .PAF A/, discrete Fourier transform .DFTA/, power spectral density .PSDA/ of A, and the definition of complex complementary sequences, we refer the reader to [4]. If we have a collection of complex complementary sequences of length v D dm, then we can compress them to obtain complementary sequences of length d. We refer to the ratio v=d D m as the compression factor. Here is the precise definition.

Deﬁnition 3.1. Let A D Œa0; a1;:::;av1 be a complex sequence of length v D dm and set .d/ ::: ; 0;:::; 1: aj D aj C ajCd C C ajC.m1/d j D d (6)

.d/ Œ .d/; .d/;:::; .d/ Then we say that the sequence A D a0 a1 ad1 is the m-compression of A. Let X; Y be an SDSs with parameters .vI r; sI / with v D 2n (and n D rCs). Assume that v D md is a nontrivial factorization. Let A; B be the binary sequences of length v associated with X and Y, respectively. Then the m-compressed sequences A.d/; B.d/ form a complementary pair. In general they are not binary sequences, their terms belong to the set fm; m 2;:::;m C 2;mg. The search for such pairs X; Y is broken into two stages: first we construct the candidate complementary sequences A.d/; B.d/ of length d, and second we lift each of them and search to find the required pairs .X; Y/. Each of the stages requires a lot of computational resources. There are additional theoretical results that can be used to speed up these computations. Some of them are described in [4], namely we use “bracelets” and “charm bracelets” to speed up the first stage. We use [5, Theorem 1] to speed up the second stage.

4 Multiplication of Golay and Periodic Golay Pairs

0 If Z Â Zv,wesetZ D Zv n Z.ToZ we associate the binary sequence Œa0; a1;:::;av1,whereai D1 if i 2 Z and ai DC1 otherwise. This gives a one-to-one correspondence between subsets Z Â Zv and the set of binary sequences Periodic Golay Pairs of Length 72 87 of length v.If.X; Y/ is an SDSs with parameters .vI r; sI / such that v D 2n, .n D r C s /, then the associated binary sequences of X and Y form a periodic Golay pair. Conversely, each periodic Golay pair of length v>1arises in this way from an SDSs with v D 2n. If there exists a Golay pair resp. a periodic Golay pair of length v,thenwesay that v is a Golay number resp. a periodic Golay number. We denote the set of Golay numbers by and the set of periodic Golay numbers by ˘.By0 we denote the set a b c of known Golay numbers, i.e., 0 Df2 10 26 W a; b; c 2 ZCg,whereZC is the set of nonnegative integers. It is not known whether 0 D . Since every Golay pair is also a periodic Golay pair, we have Â ˘. Moreover, this inclusion is strict. Indeed, the periodic Golay numbers v D 34; 50; 58; 68; 72; 74; 82 (see [6]) are not in (see [2]). If X and Y are sets of positive integers, we shall denote by XY the set of all products xy with x 2 X and y 2 Y. Given a Golay pair of length g and a periodic Golay pair of length v, then one can multiply them to obtain a periodic Golay pair of length gv. In fact there are now two such multiplications which are essentially different. Consequently, the set ˘ n 0 is infinite as it contains the set 0 f34; 50; 58; 72; 74; 82; 122; 202; 226g. The first multiplication is described in the very recent paper [8]. It is an easy consequence of [9, Theorems 13,16]. We give below a simple description in terms of the SDSs .X; Y/ associated with a periodic Golay pair. The parameters .vI r; sI / and n D r C s of this SDSs satisfy the equation v D 2n. Proposition 4.1. Let .U; V/ be a Golay pair of length g and .X; Y/ the SDS associated with a periodic Golay pair of length v D 2n. Let x; y be two indeterminates and define the sequence A D Œa0; a1;:::;av1 by setting 8 ˆ x; if i 2 X \ Y; <ˆ x; if i 2 X0 \ Y0; ai D ˆ ; ; :ˆ y if i 2 X n Y y; if i 2 Y n X:

Next, let B be the sequence obtained from A by ﬁrst reversing A and then simultaneously replacing x with y and y with x. Finally, by replacing in both A and B the indeterminates x and y with U and V, respectively, one obtains a periodic Golay pair of length gv. We observed subsequently that Turyn’s multiplication of Golay pairs provides also the multiplication of Golay and periodic Golay pairs. For convenience let us associate with each binary sequence A D Œa0; a1;:::;av1 the polynomial A.z/ WD v1 a0 Ca1zCCav1z in the indeterminate z. Then Turyn’s multiplication .A; B/ .C; D/ D .E; F/ of Golay pairs .A; B/ of length g and .C; D/ of length v is given by the formulas (see [10]) 88 D.Ž. -Dokovic´ and I.S. Kotsireas

1 1 E.z/ ŒA.z/ B.z/C.zg/ ŒA.z/ B.z/D.zg/zgvg; D 2 C C 2 (7) 1 1 F.z/ ŒB.z/ A.z/C.zg/zgvg ŒA.z/ B.z/D.zg/: D 2 C 2 C (8)

The product pair .E; F/ is a Golay pair of length gv. Proposition 4.2. If .A; B/ is a Golay pair of length g and .C; D/ is a periodic Golay pair of length v, then the pair .E; F/ given by the formulas (7) and (8) is a periodic Golay pair of length gv. Proof. The fact that .A; B/ is a Golay pair is equivalent to the identity

A.z/A.z1/ C B.z/B.z1/ D 2g: (9)

Similarly, the fact that .C; D/ is a periodic Golay pair is equivalent to the congruence

C.z/C.z1/ C D.z/D.z1/ Á 2v mod .zv 1/; (10) where .zv 1/ is the ideal of the Laurent polynomial ring ZŒz; z1 generated by zv 1. A computation gives that

4E.z/E.z1/ D .A.z/ C B.z//.A.z1/ C B.z1//C.zg/C.zg/ C .A.z/ B.z//.A.z1/ B.z1//D.zg/D.zg/ C .A.z/ C B.z//.A.z1/ B.z1//C.zg/D.zg/zggv C .A.z/ B.z//.A.z1/ C B.z1//C.zg/D.zg/zgvg;

4F.z/F.z1/ D .A.z/ B.z//.A.z1/ B.z1//C.zg/C.zg/ C .A.z/ C B.z//.A.z1/ C B.z1//D.zg/D.zg/ C .B.z/ A.z//.A.z1/ C B.z1//C.zg/D.zg/zgvg C .A.z/ C B.z//.B.z1/ A.z1//C.zg/D.zg/zggv:

By using (9) we obtain that

E.z/E.z1/ C F.z/F.z1/ D g.C.zg/C.zg/ C D.zg/D.zg//: (11)

It follows from (10)that

C.zg/C.zg/ C D.zg/D.zg/ Á 2v mod .zgv 1/; (12) and so

E.z/E.z1/ C F.z/F.z1/ Á 2gv mod .zgv 1/: (13)

This means that .E; F/ is a periodic Golay pair. Periodic Golay Pairs of Length 72 89

As an example, let us take the Golay pair .A D Œ1; 1; B D Œ1; 1/ of length g D 2 and the periodic Golay pair .C; D/ of length v D 34 with associated SDSs .X; Y/ given by

X Df0; 1; 2; 3; 5; 6; 8; 12; 13; 14; 15; 18; 20; 22; 24; 31g; Y Df0; 1; 4; 5; 7; 8; 9; 14; 15; 18; 23; 26; 28g:

Its parameters are .v D 34I r D 16; s D 13I D 12/ and n D 17. We compute the product .E; F/ D .A; B/ .C; D/ by using the multiplication from Propositions 4.1 and 4.2. The associated SDSs .P; Q/ and .R; S/, respectively, are given by

P Df0; 2; 8; 9; 10; 14; 15; 16; 18; 19; 21; 23; 28; 30; 33; 35; 36; 39; 43; 46; 47; 51; 52; 53; 55; 56; 57; 59; 61; 65; 67g; Q Df0; 1; 2; 3; 5; 6; 7; 8; 9; 10; 12; 13; 14; 16; 17; 19; 20; 23; 24; 25; 27; 28; 29; 32; 33; 34; 35; 41; 43; 44; 45; 46; 47; 48; 52; 55; 58; 61; 63g; R Df4; 5; 7; 13; 18; 21; 22; 23; 25; 26; 27; 30; 33; 35; 36; 38; 39; 40; 41; 42; 43; 45; 49; 50; 51; 54; 55; 56; 59; 60; 61; 62; 63; 64; 65; 66; 67g; S Df1; 3; 5; 7; 10; 11; 13; 14; 17; 20; 25; 27; 29; 30; 31; 36; 37; 38; 41; 45; 48; 49; 50; 52; 56; 58; 63; 64; 66g:

0 After replacing Q with its complement Q in Zv, the parameters of these two SDSs are .68I 31; 29I 26/. However, one can verify that they are not equivalent as SDSs. Indeed, the canonical forms (see [3]) .PO; QO0/ and .RO; SO/ of the SDSs .P; Q0/ and .R; S/ are given by

PO Df0; 1; 2; 3; 6; 7; 8; 10; 13; 14; 16; 17; 18; 20; 24; 28; 31; 33; 35; 36; 38; 40; 41; 43; 44; 49; 52; 53; 55; 62; 64g; QO0 Df0; 1; 2; 3; 4; 5; 11; 12; 13; 16; 18; 19; 20; 23; 24; 25; 26; 28; 30; 36; 39; 41; 42; 45; 50; 51; 55; 60; 64g; RO Df0; 1; 2; 3; 4; 6; 10; 12; 13; 14; 17; 19; 21; 23; 26; 27; 29; 32; 35; 37; 38; 41; 42; 43; 49; 51; 53; 56; 60; 61; 65g; SO Df0; 1; 3; 4; 5; 6; 8; 9; 10; 11; 13; 15; 16; 20; 23; 24; 26; 27; 28; 36; 38; 41; 44; 45; 50; 51; 52; 57; 58g:

It is rather surprising that the two multiplications described above produce nonequivalent periodic Golay pairs. 90 D.Ž. -Dokovic´ and I.S. Kotsireas

5 Computational Results for Periodic Golay Pairs

No v 2 is divisible by a prime congruent to 3 modulo 4 (see [7]). So far, none of the known members of ˘ were divisible by a prime congruent to 3 modulo 4. Hence, the periodic Golay pairs constructed below are the ﬁrst examples having the length divisible by a prime congruent to 3 modulo 4, namely the prime 3. Consequently, no periodic Golay pair of length 72 can be constructed by multiplying a nontrivial Golay pair and a periodic Golay pair. We list eight pairwise nonequivalent SDSs with parameters .72I 36; 30I 30/.As n D 36 we have v D 2n, and so these SDSs give periodic Golay pairs of length 72. All solutions are in the canonical form deﬁned in [3] and since they are different, this implies that they are pairwise nonequivalent.

.1/ f0; 1; 2; 3; 4; 5; 6; 7; 10; 12; 13; 15; 18; 20; 22; 24; 26; 27; 29; 30; 31; 35; 37; 39; 40; 43; 44; 47; 51; 52; 53; 56; 58; 59; 62; 63g; f0; 1; 2; 3; 5; 6; 8; 11; 12; 13; 14; 15; 18; 21; 23; 25; 29; 32; 33; 39; 41; 42; 43; 47; 48; 55; 56; 62; 67; 69g; .2/ f0; 1; 2; 3; 4; 5; 6; 7; 10; 12; 13; 15; 18; 20; 22; 24; 26; 27; 29; 30; 31; 35; 37; 39; 40; 43; 44; 47; 51; 52; 53; 56; 58; 59; 62; 63g; f0; 2; 3; 5; 7; 8; 9; 11; 14; 15; 17; 18; 19; 23; 24; 30; 31; 32; 33; 37; 38; 41; 42; 44; 48; 49; 51; 59; 61; 69g; .3/ f0; 1; 2; 3; 5; 7; 10; 11; 12; 13; 15; 17; 19; 20; 26; 27; 28; 29; 30; 32; 34; 35; 38; 39; 40; 42; 43; 46; 49; 51; 54; 56; 59; 60; 63; 64g; f0; 1; 2; 3; 4; 6; 7; 8; 9; 14; 15; 16; 20; 22; 24; 26; 27; 31; 33; 36; 37; 40; 42; 43; 46; 49; 54; 57; 58; 68g; .4/ f0; 1; 2; 3; 5; 7; 10; 11; 12; 13; 15; 17; 19; 20; 26; 27; 28; 29; 30; 32; 34; 35; 38; 39; 40; 42; 43; 46; 49; 51; 54; 56; 59; 60; 63; 64g; f0; 1; 3; 4; 6; 7; 8; 9; 10; 14; 15; 18; 19; 20; 22; 25; 26; 31; 32; 36; 38; 40; 42; 45; 49; 51; 52; 57; 58; 60g; .5/ f0; 1; 2; 4; 5; 6; 7; 9; 10; 11; 14; 15; 16; 17; 22; 23; 25; 26; 29; 30; 33; 35; 37; 38; 43; 45; 46; 48; 50; 51; 52; 54; 55; 60; 62; 63g; f0; 2; 3; 5; 7; 8; 9; 11; 14; 17; 18; 19; 21; 23; 24; 27; 30; 31; 32; 37; 38; 41; 42; 44; 48; 49; 57; 59; 61; 63g; .6/ f0; 1; 3; 4; 5; 6; 7; 8; 9; 10; 14; 15; 17; 18; 19; 20; 22; 25; 26; 29; 31; Periodic Golay Pairs of Length 72 91

32; 36; 38; 40; 41; 42; 45; 49; 51; 52; 53; 57; 58; 60; 65g; f0; 1; 2; 5; 7; 10; 11; 12; 13; 17; 19; 20; 26; 28; 29; 30; 32; 34; 35; 38; 40; 42; 43; 46; 49; 54; 56; 59; 60; 64g; .7/ f0; 1; 3; 4; 5; 6; 7; 8; 9; 10; 14; 15; 17; 18; 19; 20; 22; 25; 26; 29; 31; 32; 36; 38; 40; 41; 42; 45; 49; 51; 52; 53; 57; 58; 60; 65g; f0; 1; 3; 4; 5; 6; 9; 10; 13; 16; 18; 19; 21; 23; 24; 27; 30; 34; 35; 40; 46; 47; 48; 49; 53; 55; 57; 63; 65; 67g; .8/ f0; 2; 3; 4; 5; 7; 8; 9; 11; 14; 15; 16; 17; 18; 19; 23; 24; 28; 30; 31; 32; 33; 37; 38; 40; 41; 42; 44; 48; 49; 51; 52; 59; 61; 64; 69g; f0; 1; 2; 4; 5; 6; 7; 10; 12; 13; 18; 20; 22; 24; 26; 29; 30; 31; 35; 37; 40; 43; 44; 47; 52; 53; 56; 58; 59; 62g:

Let v 2 ˘ and v>1. Then it is known that v must be even and v=2 must be a sum of two squares. Moreover there is an SDSs with parameters .vI r; sI / such that v D 2n. The Arasu–Xiang condition [1, Corollary 3.6] for the existence of such SDSs must be satisﬁed. This gives another restriction on v. The product 0S,whereS Df1; 34; 50; 58; 72; 74; 82; 122; 202; 226g,istheset of lengths of the currently known periodic Golay pairs. For reader’s convenience we list the integers in the range 1

90; 106; 130; 146; 170; 178; 180; 194; 212; 218; 234; 250; 274; 290; 292; 298:

These are the smallest lengths for which the existence question of periodic Golay pairs remains unsolved.

Acknowledgements The authors wish to acknowledge generous support by NSERC. This research was enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca). We thank a referee for his suggestions.

References

1. Arasu, K.T., Xiang, Q.: On the existence of periodic complementary binary sequences. Des. Codes Crypt. 2, 257–262 (1992) 2. Borwein, P.B., Ferguson, R.A.: A complete description of Golay pairs for lengths up to 100. Math. Comput. 73(246), 967–985 (2003) 3. -Dokovic,´ D.Ž.: Cyclic .vI r; sI / difference families with two base blocks and v Ä 50. Ann. Comb. 15, 233–254 (2011) 92 D.Ž. -Dokovic´ and I.S. Kotsireas

4. -Dokovic´, D.Ž., Kotsireas, I.S.: Compression of periodic complementary sequences and applications. Des. Codes Crypt. 74, 365–377 (2015) 5. -Dokovic,´ D.Ž., Kotsireas, I.S:. D-optimal matrices of orders 118, 138, 150, 154 and 174 (to appear) 6. -Dokovic,´ D.Ž., Kotsireas, I.S.: Some new periodic Golay pairs. Numer. Algorithms (to appear). doi: 10.1007/s11075-014-9910-4 7. Eliahou, S., Kervaire, M., Saffari, B.: A new restriction on the lengths of Golay complementary sequences. J. Comb. Theory Ser. A 55, 49–59 (1990) 8. Georgiou, S.D., Stylianou, S., Drosou, K., Koukouvinos, C.: Construction of orthogonal and nearly orthogonal designs for computer experiments. Biometrika 101(3), 741–747 (2014) 9. Koukouvinos, C., Seberry, J.: New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function – a review. J. Stat. Plann. Inference 81, 153–182 (1999) 10. Turyn, R.J.: Hadamard matrices, Baumert-Hall units, four symbol sequences, puls compression and surface wave encodings. J. Comb. Theory Ser. A 16, 313–333 (1974) Classifying Cocyclic Butson Hadamard Matrices

Ronan Egan, Dane Flannery, and Padraig Ó Catháin

Dedicated to Hadi Kharaghani on the occasion of his 70th birthday

Abstract We classify all the cocyclic Butson Hadamard matrices BH.n; p/ of order n over the pth roots of unity for an odd prime p and np Ä 100. That is, we compile a list of matrices such that any cocyclic BH.n; p/ for these n, p is equivalent to exactly one element in the list. Our approach encompasses non-existence results and computational machinery for Butson and generalized Hadamard matrices that are of independent interest.

Keywords Automorphism group • Butson Hadamard matrix • Cocyclic • Rela- tive difference set

Mathematics Subject Classiﬁcation (2010): 05B20, 20B25, 20J06

1 Introduction

We present a new classiﬁcation of Butson Hadamard matrices within the framework of cocyclic design theory [9, 16]. New non-existence results are also obtained. We extend M AGMA [1]andGAP [13] procedures implemented previously for 2- cohomology and relative difference sets [12, 21, 23] to determine the matrices and sort them into equivalence classes.

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. R. Egan • D. Flannery () School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland e-mail: dane.ﬂ[email protected] P. Ó Catháin School of Mathematical Sciences, Monash University, Melbourne, VIC 3800, Australia

© Springer International Publishing Switzerland 2015 93 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_8 94 R. Egan et al.

Cocyclic development was introduced by de Launey and Horadam in the 1990s, as a way of handling pairwise combinatorial designs that exhibit a special symmetry. It has turned out to be a powerful tool in the study of real Hadamard matrices (see [21] for the most comprehensive classification). A basic strategy, which we follow here, is to use algebraic and cohomological techniques in systematically constructing the designs. Butson Hadamard matrices have applications in disparate areas such as quantum physics and error-correcting codes. So lists of these objects have value beyond design theory. We were motivated to undertake the classification in this paper as a first step towards augmenting the available data on complex Hadamard matrices (and we did find several matrices not equivalent to any of those in the online catalog [3]). Specifically, we classify all Butson Hadamard matrices of order n over pth roots of unity for an odd prime p and np Ä 100. The restriction to pth roots is a convenience that renders each matrix generalized Hadamard over a cyclic group of order p; for these we have a correspondence with central relative difference sets that enables us to push the computation to larger orders. It must be emphasized that most of the techniques that we present apply with equal validity to generalized Hadamard matrices over any abelian group—but are not valid for Butson Hadamard matrices over kth roots of unity with k composite. Moreover, the tractability of the problem considered in this paper suggests avenues for investigation of other cocyclic designs, such as complex weighing matrices and orthogonal designs. The paper is organized as follows. In Sect. 2 we set out background from design theory: key definitions, our understanding of equivalence, and general non-existence results. Section 3 is devoted to an explanation of our algorithm to check whether two Butson Hadamard matrices are equivalent. We recall the necessary essentials of cocyclic development in Sect. 4.TheninSect.5 we specialize to cocyclic Butson Hadamard matrices. The full classification is outlined in Sect. 6. We end the paper with some miscellaneous comments prompted by the classification. For space reasons, the listing of matrices in our classification is not given herein. It may be accessed at [10].

2 Background

Throughout, p is a prime and G, K are ﬁnite non-trivial groups. We write k for e2i=k.

2.1 Butson and Generalized Hadamard Matrices

A Butson Hadamard matrix of order n and phase k, denoted BH.n; k/,isann n matrix H with entries in hki such that HH D nIn over C.HereH is the usual Hermitian, i.e., complex conjugate transpose. Classifying Cocyclic Butson Hadamard Matrices 95

For n divisible by jKj,ageneralized Hadamard matrix GH.n; K/ of order n over K is an n n matrix H D Œhij whose entries hij lie in K and such that n P HH D nI C . x/.J I / n jKj x2K n n

Œ 1 1 where H D hji , Jn is the all s matrix, and the matrix operations are performed over the group ring ZK. The transpose of a BH.n; k/ is a BH.n; k/; the transpose of a GH.n; K/ is not necessarily a GH.n; K/, except when K is abelian [9, Theorem 2.10.7]. However, if H is a Butson or generalized Hadamard matrix, then so too is H. For the next couple of results, see Theorem 2.8.4 and Lemma 2.8.5 in [9](the former requires a theorem from [18]).

Theorem 2.1. If there exists a BH.n; k/, and p1;:::;pr are the primes dividing k, then n D a1p1 CCarpr for some a1;:::;ar 2 N. One consequence of Theorem 2.1 is that BH.n; pt/ can exist only if pjn. P ! n !i 0 < Lemma 2.1. Let be a primitive pth root of unity. Then iD0 ai D for n p and a0;:::;an 2 N not all zero if and only if n D p 1 and a0 DDan. P P Z ZŒ k1 i k1 i Let C DhxiŠCk and deﬁne k W C ! k by k iD0 cix D iD0 ci k. The map k extends to a ring epimorphism Mat.n; ZC/ ! Mat.n; ZŒk/.

Lemma 2.2. If M is a GH.n; Ck/,thenk.M/ is a BH.n; k/;ifMisaBH.n; p/,then 1. / . ; / p M is a GH n Cp . Proof. The ﬁrst part is easy, and the second uses Lemma 2.1. ut

Thus, a BH.n; p/ is the same design as a GH.n; Cp/. Butson’s seminal paper [4] supplies a construction of BH.2apb; p/ for 0 Ä a Ä b. Example 2.1. For composite n, the Fourier matrix (more properly, Discrete Fourier Transform matrix) of order n is a BH.n; n/ but not a GH.n; Cn/. Example 2.2. There are no known examples of GH.n; K/ when K is not a p-group. Indeed, finding a GH.n; K/ with jKjDn not a power of p would resolve a long- standing open problem in finite geometry; namely, whether a finite projective plane always has prime-power order.

2.2 Equivalence Relations

Let X, Y be GH.n; K/s. We say that X and Y are equivalent if MXN D Y for monomial matrices M, N with non-zero entries in K.IfX, Y are BH.n; k/s, then they are equivalent if MXN D Y for monomials M, N with non-zero entries from hki. Equivalence in either situation is denoted X Y, whereas if M, N are permutation 96 R. Egan et al. matrices then X, Y are permutation equivalent and we write X Y. The equivalence operations deﬁned above are local, insofar as they are applied entrywise to a single row or column one at a time. We will not regard taking the transpose or Hermitian as equivalence operations. If H is a GH.n; K/,thenH H0 where H0 is normalized (its ﬁrst row and column are all 1s) and thus row-balanced: each element of K appears with the same frequency, n=jKj, in each non-initial row. Similarly, H0 is column-balanced. Unless k is prime, neither property may hold for a normalized BH.n; k/.

2.3 Non-existence of Generalized Hadamard Matrices

Certain number-theoretic conditions exclude various odd n as the order of a generalized Hadamard matrix; see, e.g., [5, 6, 25]. The main general result of this kind that we need is due to de Launey [6]. Theorem 2.2. Let K be abelian, and r, n be odd, where r is a prime dividing jKj.If a GH.n; K/ exists, then every integer m 6Á 0 mod r that divides the square-free part of n has odd multiplicative order modulo r. Remark 2.1. BH.n; p/ do not exist for .n; p/ 2f.15; 3/; .33; 3/; .15; 5/g. We shall derive non-existence conditions for cocyclic BH.n; p/ later.

3 Deciding Equivalence of Butson Hadamard Matrices

In this section we give an algorithm to decide equivalence of Butson Hadamard matrices. The problem is reduced to deciding graph isomorphism, which we carry out using Nauty [19]; and subgroup conjugacy and intersection problems, routines for which are available in MAGMA.

3.1 Automorphism Groups, the Expanded Design, and the Associated Design

The direct product Mon.n; hki/ Mon.n; hki/ of monomial matrix groups acts on the (presumably non-empty) set of BH.n; k/ via .M; N/H D MHN. The orbit of H is its equivalence class; the stabilizer is its full automorphism group Aut.H/. Example 3.1 ([9, Section 9.2]). Denote the r-dimensional GF.p/-space by V.Then > r D D Œxy x;y2V is a GH.p ; Cp/, written additively. In fact D is the r-fold Kronecker product of the Fourier matrix of order p (so when p D 2 we get the Sylvester matrix). 1 >2 . / . r / Ì . ; / If r ¤ or p , then Aut D Š Cp Cp AGL r p . Classifying Cocyclic Butson Hadamard Matrices 97

Let Perm.n/ be the group of all n n permutation matrices. The permutation automorphism group PAut.X/ of an n n array X consists of all pairs .P; Q/ 2 Perm.n/2 such that PXQ> D X. Clearly PAut.H/ Ä Aut.H/. The array X is group- developed over a group G of order n if X Œh.xy/x;y2G for some map h. We readily prove that X is group-developed over G if and only if G is isomorphic to a regular subgroup (i.e., subgroup acting regularly in its induced actions on the sets of row and column indices) of PAut.X/. The full automorphism group Aut.H/ has no direct actions on rows or columns E ŒiCj of H. Rather, it acts on the expanded design H D k H via a certain isomorphism of Aut.H/ onto PAut.EH/:see[9, Theorem 9.6.12].

Proposition 3.1 ([9, Corollary 9.6.10]). If H1 and H2 are equivalent BH.n; k/s, then EH1 EH2 ; therefore, PAut.EH1 / and PAut.EH2 / are isomorphic as conjugate subgroups of Perm.nk/2. A converse of Proposition 3.1 also holds, which we might use as a criterion to distinguish Butson Hadamard matrices. For computational purposes it is preferable to work with the .0; 1/-matrix AH (the associated design of H) obtained from EH by setting its non-identity entries to zero. Then we need an analog of Proposition 3.1 for the associated design. Before stating this, we say a bit more about the embedding 2 2 .1/ .2/ .1/ W Mon.n; hki/ ! Perm.nk/ .Itmaps.P; Q/ to . .P/; .Q// where (resp. .2/) replaces each non-zero entry by the permutation matrix representing that entry in the right (resp. left) regular action of hki on itself. Denote the image 2 of Mon.n; hki/ under by M.n; k/.

Proposition 3.2. Let H1,H2 be BH.n; k/s. We have H1 H2 if and only if AH1 D > XAH2 Y for some .X; Y/ 2 M.n; k/. P .1/. / .2/. /> E Proof. Suppose that P AH2 Q D AH1 , and write Hi D r2h i rHi;r (so / k AHi D Hi;1 . By Theorem 9.6.7 and Lemma 9.8.3 of [9],

.1/ .2/ > H1;r D .P/H2;r .Q/ :

Therefore EH1 D EPH2Q by [9, Lemma 9.6.8]. This implies that H1 D PH2Q . ut We also use the following simple fact. Lemma 3.1. Let A, B be subgroups and x, y be elements of a group G. Then either xA \ yB D;,orxA\ yB D g.A \ B/ for some g 2 G. We now state our algorithm to decide equivalence of Butson Hadamard matrices H1 and H2 of order n and phase k.

1. Compute G1 D PAut.AH1 / with Nauty. 2 2. Attempt to ﬁnd 2 Perm.nk/ such that AH1 D AH2 . If no such exists then return false. 3. Compute U D G1 \ M.n; k/ and a transversal T for U in G1. 4. If there exists t 2 T such that t 2 M.n; k/ then return true; else return false. 98 R. Egan et al.

If H1 H2,thenG1 \ M.n; k/ ¤;by Proposition 3.2,sobyLemma3.1 we must ﬁnd a t as in step 4. A report of false is then correct by Proposition 3.2; a report of true is clearly correct. Note that if the algorithm returns true then we ﬁnd an 1 element .t/ mapping H1 to H2. Step 1 is a potential bottleneck, although it remains feasible for graphs with several hundred vertices. Equivalence testing is therefore practicable for many BH.n; k/ that have been considered in the literature. Example 3.2. The authors of [20] construct a series of BH.2p; p/ but cannot decide whether their matrices are equivalent to those of Butson [4, Theorem 3.5]. Our method, which has been implemented in MAGMA, shows that the BH.10; 5/ denoted S10 in [20] is equivalent to Butson’s matrix in less than 0:1s (an explicit equivalence is given at [10]).

4 Cocyclic Development

Since our main concern is Butson Hadamard matrices, we recap the essential ideas of cocyclic development solely for this type of design.

4.1 Second Cohomology and Designs

Let H be a BH.n; k/,andletW be the k k block circulant matrix with first row .0n;:::;0n; In/. A regular subgroup of PAut.EH/ containing the central element > .W ; W/ is centrally regular.By[9, Theorem 14.7.1], PAut.EH/ has a centrally regular subgroup if and only if H Œ .x; y/x;y2G for some G and cocycle W G G !hki; i.e., .x; y/ .xy; z/ D .x; yz/ .y; z/ 8 x; y; z 2 G.We say that H Œ .x; y/x;y2G is cocyclic, with indexing group G and cocycle .A cocycle of H is orthogonal. Let U be a finite abelian group and denote the group of all cocycles W GG ! U by Z.G; U/. Our cocycles are normalized, meaning that .x; y/ D 1 when x or y is 1.If W G ! U is a normalized map, then @ 2 Z.G; U/ defined by @ .x; y/ D .x/1 .y/1 .xy/ is a coboundary. These form a subgroup B.G; U/ of Z.G; U/,andH.G; U/ D Z.G; U/=B.G; U/ is the second cohomology group of G. For each 2 Z.G; U/, the central extension E. / of U by G is the group with elements f.g; u/ j g 2 G; u 2 Ug and multiplication given by .g1; u1/.g2; u2/ D .g1g2; u1u2 .g1; g2//.Conversely,letE be a central extension of U by G, with embedding W U ! E and epimorphism W E ! G satisfying ker D .U/. Choose a normalized map W G ! E so that D idG.Then .x; y/ D 1 1 . .x/ .y/ .xy/ / defines a cocycle ,andE. / Š E. Different choices of right inverse of do not alter the cohomology class of . Classifying Cocyclic Butson Hadamard Matrices 99

ABH.n; k/, H, is cocyclic with cocycle if and only if E. / is isomorphic to a centrally regular subgroup of PAut.EH/ by an isomorphism mapping .1; k/ to .W>; W/.IfH is group-developed over G,thenH is equivalent to a cocyclic BH.n; k/ with cocycle 2 B.G; hki/ and extension group E. / Š G Ck. Example 4.1. The Butson Hadamard matrix D in Example 3.1 is cocyclic, with r . r ; / . ; / > indexing group Cp and cocycle 62 B Cp Cp deﬁned by x y D xy .Note . / rC1 that is multiplicative and symmetric. If p is odd, then E Š Cp .The determination of all cocycles, indexing groups, and extension groups of D would be an interesting exercise; cf. the account for p D 2 in [9, Chapter 21].

4.2 Computing Cocycles

We compute Z.G; hki/ by means of the Universal Coefﬁcient Theorem:

H.G; U/ D I.G; U/=B.G; U/ T.G; U/=B.G; U/ where I.G; U/=B.G; U/ is the isomorphic image under inflation of Ext.G=G0; U/ 0 and T.G; U/=B.G; U/ Š Hom.H2.G/; U/.HereG D ŒG; G and H2.G/ is the Schur multiplier of G. . ; / We describe theQ calculation of I G U for U DhuiŠCp as this is used in a 0 = 0 0 ei later proof. Let ihgiG i be the Sylow p-subgroup of G G ,wherejgiG jDp . ei ei Define Mi to be the p p matrix whose rth row is .1;:::;1;u;:::;u/,thefirstu ei occurring in column p rC2.LetNi be the jGjjGj matrix obtained by taking the Kronecker product of Mi with the all 1s matrix. Up to permutation equivalence, the Ni constitute a complete set of representatives for the elements of I.G; U/=B.G; U/ displayed as cocyclic matrices. For more detail see [12].

4.3 Shift Action

In a search for orthogonal elements of Z.G; Cp/, it is not enough to test a single 0 from each cohomology class Œ 2 H.G; Cp/:if is orthogonal, then 2 Œ need not be orthogonal. Horadam [16, Chapter 8] discovered an action of G on each Œ that preserves orthogonality, deﬁned by g D @. g/ where g.x/ D .g; x/. This ‘shift’ action induces a linear representation G ! GL.V/ where V is any G- invariant subgroup of Z.G; Cp/, allowing effective computation of orbits in V [11]. 100 R. Egan et al.

4.4 Further Equivalences for Cocyclic Matrices

Equivalence operations preserving cocycle orthogonality, apart from local ones, arise from the shift action or natural actions on Z.G; hpi/ by Aut.G/Aut.Cp/.The actionbyAut.Cp/ alone furnishes a global equivalence operation. Together with the local operations these generate the holomorph CpÌCp1 of hpi [9, Theorem 4.4.10].

4.5 Central Relative Difference Sets

Theorem 4.1. There exists a cocyclic BH.n; p/ with cocycle if and only if there is a relative difference set in E. / with parameters .n; p; n; n=p/ and central forbidden subgroup h.1; p/i. Proof. This follows from [9, Corollary 15.4.2] or [22, Theorem 4.1]. ut We explain one direction of the correspondence in Theorem 4.1.LetE be a central extension of U Š Cp by G.Say embeds U into the center of E,and W E ! G is an epimorphism with kernel .U/. Suppose that R Dfd1 D 1; d2;:::;dngÂE is an .n; p; n; n=p/-relative difference set with forbidden subgroup U;i.e.,the 1 . / multiset of quotients didj for j ¤ i contains each element of E n U exactly n=p times, and contains no element of .U/.SinceR is a transversal for the cosets of .U/ in E,wehaveG Dfgi WD .di/ j 1 Ä i Ä ng.Put .gi/ D di.Then Œ .x; y/x;y2G is a BH.n; p/.

5 Cocyclic Butson Hadamard Matrices

Theorem 5.1. Let K be abelian, n DjGj be divisible by jKj, 2 Z.G; K/, and H D Œ .x; y/x;y2G.ThenHisaGH.n; K/ if and only if it is row-balanced. In that event H is column-balanced too. Proof. This follows from [16, Lemma 6.6], which generalizes a phenomenon observed for cocyclic Hadamard matrices [9, Theorem 16.2.1]. ut So we begin our classiﬁcation by searching for balanced cocycles in the relevant Z.G; Cp/.Whenk is not prime, a cocyclic BH.n; k/ need not be balanced; by [16, Lemma 6.6] again, Œ .x; y/x;y2G for 2 Z.G; hki/ is a BH.n; k/ if and only if each non-initial row sum is zero. We mention extra pertinent facts about Fourier matrices. Lemma 5.1. The Fourier matrix of order n is a cocyclic BH.n; n/ with indexing group Cn. If n is odd then it is equivalent to a group-developed matrix. Classifying Cocyclic Butson Hadamard Matrices 101

Proposition 5.1 ([14]). Every circulant BH.p; p/ is equivalent to the Fourier matrix of order p. Proposition 5.2 ([15]). For p Ä 17, the Fourier matrix of order p is the unique BH.p; p/ up to equivalence.

5.1 Non-existence of Cocyclic Butson Hadamard Matrices

As we expect, there are restrictions on the order of a group-developed Butson Hadamard matrix. .j / .j / . ; / Lemma 5.2. Set rj D Re k and sj D Im k .ABH n k with constant row and column sums exists only if there are x0;:::;xk1 2f0;1;:::;ng satisfying P P k1 2 k1 2 jD0 rjxj C jD0 sjxj D n (1) P k1 and jD0 xj D n. Proof.P Let H be a BH.n; k/ with every row and column summing to s D k1 j jD0 xj k D a C bi.Then nJn D JnHH D sJnH D ssJn implies n D a2 C b2,whichis(1). ut Remark 5.1. If k D 2,then(1) just gives that n must be square, which is well- known. If k D 4,thenn is the sum of two integer squares. As a sample of other exclusions, the following cannot be the order of a group-developed BH.n; k/. (i) k D 3, n Ä 100: 6, 15, 18, 24, 30, 33, 42, 45, 51, 54, 60, 66, 69, 72, 78, 87, 90, 96, 99. (ii) k D 5, n Ä 25: 10, 15. Some of these orders are covered by general results (see Remark 2.1). Henceforth p is odd. Lemma 5.3. Let k D pt and n D prmwherep− m. Suppose that H is a cocyclic BH.n; k/ with indexing group G such that G=G0 has a cyclic subgroup of order pr. p Then any cocycle 2 I.G; Ck/ of H is in I.G; Ck/ .

Proof. (Cf. [16, Corollary 7.44]) By Sect. 4.2,wehave D 1@ for some 0 p 1 inﬂated from Z.G=G ; Ck/ and map . Assume that 1 62 I.G; Ck/ .Then Œ 1.x; y/x;y2G has a row with m occurrences of k and every other entry equal to 1. Label this row a.Now Q Q Q Q @ . ; / . /1 . /1 . / y2G a y D y2G a y2G y y2G ay . /n p : D a 2h k i 102 R. Egan et al.

Œ . ; / So, if we multiply along row a of x y x;y2G, thenP we get an element of p k1 i 0 h kinh k i. But this is a contradiction.P For supposeP that iD0 ci k D . Since the p1 . t1/ k1 i p i 1 kth cyclotomic polynomial iD0 x divides iD0 cix ,wehavecj D cpt Cj D t1 D c.p1/pt1Cj, 0 Ä j Ä p 1. It is then straightforward to verify that Q 1 k ici p iD0 k 2h k i. ut Corollary 5.1. If n is p-square-free, then a cocyclic BH.n; p/ is equivalent to a group-developed matrix. Proof. Let G be the indexing group of a cocyclic BH.n; p/. Either p divides jG0j or Lemma 5.3 applies, and thus I.G; Cp/ D B.G; Cp/.AlsoHom.H2.G/; Cp/ D 1 by [17, Theorem 2.1.5]. ut Proposition 5.1 then yields Corollary 5.2. A cocyclic BH.p; p/ is equivalent to the Fourier matrix of order p. Remark 5.2. By Remark 5.1 and Corollary 5.1,for.n; p/ D .10; 5/ or p D 3 and n 2f6; 24; 30g, there are no cocyclic BH.n; p/ at all (so Butson’s construction [4] is not cocyclic). Furthermore, a cocyclic BH.12; 3/,BH.21; 3/,BH.20; 5/,or BH.14; 7/ is equivalent to a group-developed matrix.

5.2 Existence of Cocyclic BH.n; p/,npÄ 100

Table 1 summarizes existence of matrices in our classiﬁcation. Remark 5.3. There are non-cocyclic BH.6; 3/ and BH.10; 5/ by [4]. Non-existence of cocyclic BH.6; 3/ is established by computer in [16, Example 7.4.2]. We relied on computation of relative difference sets only for parameter values that we could not settle otherwise. Nevertheless, those calculations were not onerous. The search for a relative difference set with parameters .14; 7; 14; 2/ ran in

Table 1 Existence of BH.n; p/ n 1 2 3 4 5 6 7 8 9 10 11 p n p 3 F NC E E N S2 S1 NC E NC N 5 F NC N S1 7 F S1

N: No Butson Hadamard matrices by Remark 2.1 NC: No cocyclic Butson Hadamard matrices by Remark 5.2 E: Cocyclic Butson Hadamard matrices exist. See Sect. 6 S1: No cocyclic Butson Hadamard matrices according to a relative difference set search S2: No cocyclic Butson Hadamard matrices according to an orthogonal cocycle search F: The Fourier matrix is the only Butson Hadamard matrix by Proposition 5.2 (or Corollary 5.2) Classifying Cocyclic Butson Hadamard Matrices 103 under an hour; the test for an RDS.20; 5; 20; 4/ took about a day, with most of the time being spent on C100. We note additionally that there are theoretical obstructions to the existence of an RDS.21; 3; 21; 7/: the system of diophantine signature equations that such a difference set must satisfy does not admit a solution [24].

6 The Full Classiﬁcation

The only cases left to deal with are .n; p/ 2f.9; 3/; .12; 3/; .27; 3/g. In this section we discuss our complete and irredundant classification of such BH.n; p/. Our overall task splits into two steps. We first compute a set of cocyclic BH.n; p/ containing representatives of every equivalence class. Then we test equivalence of the matrices produced. Since our method for the second step was given in Sect. 3, and the orders involved pose no computational difficulties, we say nothing further about this step. Two complementary methods were used for the first step: checking shift orbits for orthogonal cocycles, and constructing relative difference sets. See Sects. 4.2 and 4.3; also, we refer to [21, Section 6], which discusses a classification of cocyclic Hadamard matrices via central relative difference sets. The algorithm for constructing the difference sets in this paper is identical to the one there, and was likewise carried out using Röder’s GAP package RDS [23].

Example 6.1. Table 2 lists the number t of orthogonal elements of Z.G; C3/ for jGjD9 or 12.

Table 2 Counting orthogonal elements of Z.G; C3/ 2 2 G C9 C3 C12 C3 Ì C4 Alt.4/ D6 C2 C3 t 18 144 0 288 48 0 96

If jGj2f6; 15; 18g,thent D 0.

6.1 BH.9; 3/

There are precisely three equivalence classes of cocyclic BH.9; 3/. 2 One class contains BH.3; 3/ ˝ BH.3; 3/, which has indexing group C3 and cocycle that is not a coboundary. Some matrices H1 in this class are group-developed 2 over C3.NoH1 has indexing group C9. See Examples 3.1 and 4.1. Another equivalence class contains group-developed matrices with indexing 2 group C9.NomatrixH2 in this class has indexing group C3; hence, the cocycles of H2 are all coboundaries by Lemma 5.3. This class is not represented in [3], but happens to be an example of the construction in [7](cf.[2]). A representative is the 2 2 circulant with ﬁrst row .1;1;1;1;3;3 ;1;3 ;3/. 104 R. Egan et al.

The third class contains matrices H3 H2 that are cocyclic with indexing group 2 C9.Again,H3 is equivalent to a circulant, does not have indexing group C3,allof its cocycles are coboundaries, and it is not in [3].

By Proposition 3.1,PAut.EH2 / Š PAut.EH3 /. These groups are solvable. We described PAut.EH1 / in Example 3.1.

6.2 BH.12; 3/

Each cocyclic BH.12; 3/ is equivalent to a group-developed matrix (Remark 5.2) 2 2 over one of C3 Ì C4,C2 Ì C3,orC2 C3. There are just two equivalence classes, which form a Hermitian pair. The automorphism groups have order 864. This is the only order n in our classiﬁcation which is not a prime power and for which cocyclic BH.n; p/ exist.

6.3 BH.27;3/

Predictably, order 27 was the most challenging one that we faced in our computations. An exhaustive search for orthogonal cocycles was not possible, so this order was classiﬁed by the central relative difference sets method. There are 16 equivalence classes of cocyclic BH.27; 3/ in total. Some are Kronecker products of cocyclic BH.9; 3/ with the unique BH.3; 3/, but the majority are not of this form. Each matrix is equivalent to its transpose. There are two classes that are self-equivalent under the Hermitian; the rest occur in distinct Hermitian pairs. Except for the generalized Sylvester matrix, whose automorphism group as stated in Example 3.1 is not solvable, the automorphism group of a BH.27; 3/ has order 2a3b. Every non-cyclic group of order 27 is an indexing group of at least one BH.27; 3/. There are no circulants.

7 Concluding Comments

It is noteworthy that all matrices in our classiﬁcation are equivalent to group- developed ones (non-trivial cohomology classes appear too). This may be compared with [21], which features many equivalence classes not containing group-developed Hadamard matrices. Also, while there exist circulant BH.pr; p/ for all odd p and r Ä 2 [2, 7], we have not yet found a circulant BH.n; p/ when n is not a p-power. Classifying Cocyclic Butson Hadamard Matrices 105

A few composition results should be given. Let i 2 Z.Gi; Ck/ for i D 1, 2,and deﬁne 2 Z.G1 G2; Ck/ by ..a; b/; .x; y// D 1.a; x/ 2.b; y/. It is not hard to show that 2 B.G1 G2; Ck/ if and only if 1, 2 are coboundaries.

Lemma 7.1. Suppose that Hi is a cocyclic BH.ni; k/ with cocycle i, 1 Ä i Ä 2. Then H1 ˝ H2 is a cocyclic BH.n1n2; k/ with cocycle . 2 2 Corollary 7.1. For a 1,b a, and G 2fC3 Ì C4; C2 Ì C3; C2 C3g, there exists 2a b a ba a group-developed BH.2 3 ;3/with indexing group G C3 . Corollary 7.1 was proved by de Launey [8, Corollary 3.10], albeit only for 2a b indexing groups C2 C3.

Acknowledgements R. Egan received funding from the Irish Research Council (Government of Ireland Postgraduate Scholarship). P. Ó Catháin was supported by Australian Research Council grant DP120103067.

References

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Ebrahim Ghaderpour

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract Craigen introduced and studied signed group Hadamard matrices extensively, following Craigen’s lead, studied and provided a better estimate for the asymptotic existence of signed group Hadamard matrices and consequently improved the asymptotic existence of Hadamard matrices. In this paper, we introduce and study signed group orthogonal designs (SODs). The main results include a method for ﬁnding SODs for any k-tuple of positive integer and then an application to obtain orthogonal designs from SODs, namely, for any k-tuple u1; u2;:::;uk of positive integers, we show that there is an integer N D N.u1; u2;:::;uk/ such that for each n N, a full orthogonal design (no n n n zero entries) of type 2 u1;2 u2;:::;2 uk exists.

Keywords Asymptotic existence • Circulant matrix • Hadamard matrix • Orthogonal design • Signed group

1 Introduction

A signed group S (see [1, 2]) is a group with a distinguished central element, an element that commutes with all elements of the group, of order two. Denote the unit of a group as 1 and the distinguished central element of order two as 1.In every signed group, the set f1; 1g ˝is a normal˛ subgroup, and we call the number of elements in the quotient group S= 1 the order of signed group S: So, a signed group of order n is a group of order 2n: A signed group T is called a signed subgroup

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. E. Ghaderpour () Department of Earth and Space Science and Engineering, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3 e-mail: [email protected]

© Springer International Publishing Switzerland 2015 107 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_9 108 E. Ghaderpour of a signed group S; if T is a subgroup of S and the distinguished central elements of S and T coincide. We denote this relation by T Ä S. Example 1.1. There are a number of signed groups with different applications. We present some of them used in this work:

.i/ The trivial signed group SR Df1; ˝ 1g is a signed˛ group of order 1. 2 .ii/ The complex signed group SC D iI i D1 Df˙1; ˙ig is a signed group of order 2. ˝ ˛ ˚ 2 2 .iii/ The Quaternion« signed group SQ D j; kI j D k D1; jk Dkj D ˙ 1; ˙j; ˙k; ˙jk is a signed group of order 4. .iv/ The set of all monomial f0; ˙1g-matrices of order n, SPn, forms a group of order 2nnŠ and a signed group of order 2n1nŠ: Let S and T be two signed groups. A signed group homomorphism W S ! T is a map such that for all a; b 2 S, .ab/ D .a/ .b/ and .1/ D1.Aremrep (real monomial representation) is a signed group homomorphism W S ! SPn.A faithful remrep is a one to one remrep. 1 ; Let R be a ring with unit R and˚ P let S be a signed group« with distinguished 1 : Œ n ; central element S Then R S WD iD1 risiI ri 2 R si 2 P ˝ is the˛ signed group ring, where P is a set of coset representatives of S modulus 1S and for r 2 R; s 2 P, we make the identification rs D r.s/. Addition is defined termwise, and multiplication is defined by linear extension. For instance, r1s1.r2s2 C r3s3/ D 1; 2; 3 r1r2s1s2 C r1r3s1s3,whereri 2 R and si 2 P, i 2f g. P R RŒ : n In this work, we choose R D . Suppose x 2 S Then x D iD1Prisi,where R; n 1 ri 2 si 2 P.Theconjugation of x, denoted x,isdefinedasx WD iD1 risi . Clearly, the conjugation is an involution, i.e., x D x for all x 2 RŒS; and xy DNyxN p p p for all x; y 2 RŒS: As an example, 2j C 3jk D 2j1 C 3.jk/1 D 2j 3jk; where j; k 2 SQ: For an m n matrix A D Œaij with entries in RŒS define its adjoint as an n m t matrix A D A ˚D Œaji: Let S be a« signed group, and let A D Œaij be a square matrix such that aij 2 0; 1x1;:::;kxk ; where ` 2 S and x` is a variable, 1 Ä ` Ä k: For each aij D `x` or 0, let aij D `x` or 0, and jaijjDj`x`jDx` or 0. We define abs.A/ WD jaijj : We call A quasisymmetric,ifabs.A/ D abs.A /; where A D Œaji: Also, A is called˚ normal if AA D A A: The support« of A (see [2]) is defined by supp.A/ WD positions of all nonzero entries ofA . Suppose A D˚ a1; a2;:::;an «and B D b1; b2;:::;bn are two sequences with elements from 0; 1x1;:::;kxk ; where the xk’s are variables and k 2 S .1 Ä k Ä / n for some signed group S.WeuseAR to denote the sequence whose elements are ;:::; ; : those of A, conjugated and in reverse order (see [4]), i.e., AR D an a2 a1 We . / . /: say A is quasireverse to B if abs AR D abs B ; ;:::; A circulant matrixP C D circ a1 a2 an (see [6, chap. 4]) can be written n1 k; 0; 1; 0; : : : ; 0 : as C D a1In C kD1 akC1U where U D circ Therefore, any two circulant matrices of order n with commuting entries commute. If C D circ a1; a2;:::;an ; then C D circ a1; an;:::;a2 . Signed Group Orthogonal Designs and Their Applications 109

We use the notation u.k/ to show u repeats k times. Suppose A and B are sequences of length n such that A is quasireverse to B.LetD D circ 0.aC1/; A;0.2bC1/; B;0.a/ , where a and b are nonnegative integers and let m D 2a C 2b C 2n C 2.Then 0 ; ;0 ; ;0 . / . /: D D circ .aC1/ BR .2bC1/ AR .a/ and abs D D abs D Hence, D is a quasisymmetric circulant matrix of order m. The non-periodic autocorrelation function [9] of a sequence A D .x1;:::;xn/ of commuting square complex matrices of order m is deﬁned by 8 <ˆ Xnj 0;1;2;:::; 1 xiCjxi if j D n NA.j/ WD :ˆ iD1 0 j n

: ; ;:::; where xi denotes the conjugate transpose of xi AsetfA1 A2 A`g of sequences (not necessarilyP in the same length) is said to have zero autocorrelation if for >0; ` . / 0: all j kD1 NAk j D Sequences having zero autocorrelation are called complementary. Apair.AI B/ of f˙1g-complementary sequences of length n is called a Golay pair of length n,andapair.A1I B1/ of f˙x; ˙yg-complementary sequences of length n1 is called a Golay pair in two variables x and y of length n1. The length n is called Golay number. Similarly, a pair .CI D/ of f˙1; ˙ig-complementary sequences of length m is called a complex Golay pair of length m,andapair.C1I D1/ of f˙x; ˙ix; ˙y; ˙iyg-complementary sequences of length m1 is called a complex Golay pair in two variables x and y of length m1. The length m is called complex Golay number. In this paper, the sequences A1 and C1 are assumed to be quasireverse to B1 and D1, respectively. Craigen et al. [4] showed that if g1 and g2 are complex Golay numbers and g is an even Golay number, then gg1g2 is a complex Golay number. Using this, they showed the following theorem. Theorem 1.1. All numbers of the form m D 2aCu3b5c11d13e are complex Golay numbers, where a; b; c; d; e, and u are non-negative integers such that bCcCdCe Ä a C 2u C 1 and u Ä c C e: The following lemma is immediate from the deﬁnition of complex Golay pair. Lemma 1.1. Let .AI B/ be a complex Golay pair of length m. Then .xA; yB/; .yA; xB/ is a complex Golay pair of length 2m in two variables x and y. From Theorem 1.1 and Lemma 1.1, we have the following result. Corollary 1.1. There is a complex Golay pair in two variables of length n D 2aCuC13b5c11d13e,wherea; b; c; d; e, and u are non-negative integers such that b C c C d C e Ä a C 2u C 1 and u Ä c C e: In Sect. 2, we introduce signed group orthogonal designs (SODs), and will show some of their properties. Then as one of their applications, in Theorem 3.1,weshow how to obtain orthogonal designs from SODs. In Sect. 4, using SODs, we prove 110 E. Ghaderpour

Theorems 4.2 and 4.3 that give two different bounds for the asymptotic existence of orthogonal designs, namely, for any k-tuple .u1; u2;:::;uk/ of positive integers, there is an integer N D N.u1; u2;:::;uk/ such that a full orthogonal design of type n n n 2 u1;2 u2;:::;2 uk exists for each n N. This is an alternative approach to the results obtained in [8]. Ä Ä Ä 01 10 01 In this paper, P WD ; Q WD ; R WD and I is the identity 10 0 0 d matrix of order d,where is 1.

2 SODs and Some of Their Properties A signed group orthogonal design, SOD, of type u1;:::;uk ; where u1;:::;uk are positive integers, and of order n; is a square matrix X of order n with entries from f0; 1x1;:::;kxkg; where the xi’s are variables and j 2 S;1Ä j Ä k; for some signed group S, that satisﬁes ! Xk 2 : XX D uixi In iD1 We denote it by SOD nI u1;:::;uk : Equating all variables to 1 in any SOD of order n results in a signed group weighing matrix of order n and weight w which is denoted by SW.n; w/,where w is the number of nonzero entries in each row (column) of the SOD. We call an SOD with no zero entries a full SOD. Equating all variables to 1 in any full SOD of order n results in a signed group Hadamard matrix of order n which is denoted by SH.n; S/: Craigen [2] proved the following fundamental theorem to demonstrate a novel method for the asymptotic existence of signed group Hadamard matrices and consequently Hadamard matrices. Theorem 2.1. Let p be an odd positive integer. Then a circulant SH 2p; SP22N.p/1 exists.

Remark 2.1. An SOD over the Quaternion signed group SQ is called a Quaternion orthogonal design, QOD. An SOD over the complex signed group SC is called a complex orthogonal design, COD. An SOD over the trivial signed group SR is called an orthogonal design, OD. Lemma 2.1. Every SW.n; w/ over a ﬁnite signed group is normal. Proof. Suppose that WW D wIn; where the entries in W belong to a signed group S of order m: We show that WW D WW. The space of all square matrices of Signed Group Orthogonal Designs and Their Applications 111 order n with entries in RŒS has the standard basis with mn2 elements over the ﬁeld R. Thus, there exists an integer u such that

2 u c1W C c2W CCcuW D 0; where cu ¤ 0,andci 2 R .1 Ä i Ä u/: Multiplying the above equality from the right by .W/u1;

u2 2 u3 u1 c1w.W / C c2w .W / CCcuw W D 0:

Hence W is a polynomial in W,andsoWW D WW: ut Theorem 2.2. A necessary and sufﬁcient condition an SOD nI u1;:::;uk to exist over a signed group S is that there exists a family fA1;:::;Akg of pairwise disjoint square matrices of order n with entries from f0; Sg satisfying

;1; AiAi D uiIn Ä i Ä k (1) ;1 : AiAj DAjAi Ä i ¤ j Ä k (2) Proof. Suppose that there is a A D SOD nI u1;:::;uk over a signed group S: One can write

Xk A D xmAm; (3) mD1 where the Ai’s are square matrices of order n with entries from f0; Sg: Since the entries in A are linear monomials in the xi; the Ai’s are disjoint. Since A is an SOD, ! Xk 2 ; AA D uixi In (4) iD1 and so by using (3), ! Xk Xk Xk Xk 2 . / 2 : xmAmAm C xixj AiAj C AjAi D uixi In (5) mD1 iD1 jDiC1 iD1

In the above equality, for each 1 Ä i Ä k; let xi D 1 and xj D 0 for all 1 Ä j Ä k and j ¤ i; to get (1) and therefore (2). On the other hand, if fA1;:::;Akg are pairwise disjoint square matrices of order n with entries from f0; Sg which satisfy (1)and(2), then the left-hand side of the equality (5)givesus(4). ut 112 E. Ghaderpour

Remark 2.2. Equation (4) implies Eqs. (1)and(2). Multiply (2) from the left by Ai 1 : and then from the right by Ai to get Aj Ai DAi Aj for Ä i ¤ j Ä k Therefore, by Lemma 2.1, ! Xk Xk Xk Xk 2 . / 2 : A A D xmAmAm C xixj Ai Aj C Aj Ai D uixi In mD1 iD1 jDiC1 iD1

Thus, AA D AA: It means that every SOD over a finite signed group is normal. Lemma 2.2. There does not exist any full SOD of order n >1;if n is odd. Proof. Assume that there is a full SOD of order n >1over a signed group . ; / Œ n : S. Equating all variables to 1 in the SOD, one obtains an SH n S D hij i;jD1 One may multiply each column of the SH.n; S/; from the right, by the inverse of corresponding entry of its first row, h1j; to get an equivalent SH.n; S/ with the first row all 1 (see [2, 4] for the definition of equivalence). By orthogonality of the rows of the SH.n; S/, the number of occurrences of a given element s 2 S in each subsequent row must be equal to the number of occurrences of s. Therefore, n has to be even. ut

3 Some Applications of SODs

In this section, we adapt the methods of Livinskyi [13] to obtain generalizations and improvements of his results about Hadamard matrices in the much more general setting of ODs. Suppose that we have a remrep W S ! SPm:We extend this remrep to a ring homomorphism W RŒS ! MmŒR linearly by r1s1 CCrnsn D r1 s1 C 1 t C rn sn : Since for every matrix A 2 SPm we have A D A ,foreverys 2 S, s D .s/1 D .s/t: Next theorem shows how one can obtain ODs from SODs. Theorem 3.1. Suppose that there exists an SOD nI u1;:::;uk for some signed group S equipped with a remrep of degree m; where m is the order of a Hadamard matrix. Then there is an OD mnI mu1;:::;muk : Proof. Suppose that there exists an SOD nI u1;:::;uk for some signed group S: By Theorem 2.2, there are pairwise disjoint matrices A1;:::;Ak of order n with entries in f0; Sg such that

A˛A˛ D u˛In;1Ä ˛ Ä k; (6) A˛Aˇ DAˇA˛;1Ä ˛ ¤ ˇ Ä k: (7) Signed Group Orthogonal Designs and Their Applications 113

Let W S ! SPm be a remrep of degree m; and H be a Hadamard matrix of degree m: Also, for each 1 Ä ˛ Ä k; let h i n B˛ D A˛Œi; j H : i:jD1

By Proposition 1.1 in [6], it is sufﬁcient to show that B˛’s are pairwise disjoint matrices of order mn; with f0; ˙1g entries such that

t B˛B˛ D mu˛Imn;1Ä ˛ Ä k; (8) t t B˛Bˇ DBˇB˛;1Ä ˛ ¤ ˇ Ä k: (9)

Since A˛’s are pairwise disjoint, so are B˛’s (see [6, chap. 1] for Hurwitz-Radon matrices and their properties). Let 1 Ä ˛ ¤ ˇ Ä k and 1 Ä i; j Ä n: Then Xn t t t B˛Bˇ Œi; j D A˛Œi; k HH AˇŒj; k kD1 Xn D m A˛Œi; k AˇŒj; k kD1 Xn Á D m A˛Œi; kAˇŒj; k kD1 Á D m A˛Aˇ Œi; j (10) Á D m AˇA˛ Œi; j from (7) Á Dm AˇA˛ Œi; j (11)

On the other hand, similarly, Xn t t t BˇB˛ Œi; j D AˇŒi; k HH A˛Œj; k kD1 Xn D m AˇŒi; k A˛Œj; k kD1 Xn Á D m AˇŒi; kA˛Œj; k kD1 Á D m AˇA˛ Œi; j : (12) 114 E. Ghaderpour

Comparing (11)and(12), one obtains (9). If ˛ D ˇ in (10), then for 1 Ä i; j Ä n; Á t B˛B˛ Œi; j D m A˛A˛ Œi; j D m iju˛ 1S from (6)

D miju˛Im; where ij D 1 if i D j; and 0 otherwise. Whence (8) follows. ut In the following two corollaries, it is shown how to obtain ODs from CODs and QODs. Corollary 3.1. If there exists a COD nI u1;:::;uk ; then an OD 2nI 2u1;:::;2uk exists. Proof. A COD nI u1;:::;uk can be viewed as an SOD nI u1;:::;uk over the complex signed group SC: It can be seen that W SC ! SP2 deﬁned by Ä 01 i ! R D 0 is a remrep of degree 2, and so by Theorem 3.1, there exists an OD 2nI 2u1;:::;2uk : ut Corollary 3.2. If there exists a QOD nI u1;:::;uk ; then an OD 4nI 4u1;:::;4uk exists. Proof. A QOD nI u1;:::;uk can be viewed as an SOD nI u1;:::;uk over the Quaternion signed group SQ: It can be seen that W SQ ! SP4 deﬁned by 2 3 2 3 0010 0001 6 7 6 7 6 00017 6 00 0 7 j ! R ˝ I2 D 4 5 and k ! P ˝ R D 4 5; 000 0100 0 00 000 is a remrep of degree 4, and so by Theorem 3.1, there exists an OD 4nI 4u1;:::;4uk : ut Following similar techniques in [2, 3, 13], we have the following Lemma. Lemma 3.1. Suppose˚ that A and B« are two disjoint circulant matrices of order d with entries from 0; 1x1;:::;kxk ; where the x`’s are variables, ` 2 S .1 Ä ` Ä k/ for A and ` 2 Z.S/, the center of S, .1 Ä ` Ä k/ for B: Also, assume A is normal. If Ä A C BA B C D ; A B A B Signed Group Orthogonal Designs and Their Applications 115

then CC D C C D 2I2 ˝ .AA C BB /: Moreover, if A and B are both quasisymmetric and S has a faithful remrep of degree m, then˚ there exists a circulant« 0; 0 ;:::;0 quasisymmetric normal matrix D of order d with entries from 1x1 kxk and 0 0 the same support as ACB such that DD D AA CBB ; where ` 2 S .1 Ä ` Ä k/; and S0 S is a signed group having a faithful remrep of degree 2m. Proof. It may be verified directly that CC D C C D 2I2 ˝ .AA C BB /: To find matrix D; first reorder the rows and columns of C to get matrix D0 which is a partitioned matrix of order 2d into 2 2 blocks whose entries are the .i; j/, .i Cd; j/, .i; j C d/,and.i C d; j C d/ entries of C;1Ä i; j Ä d: Applying the same reordering to 2I2 ˝.AA CBB /; one obtains .AA CBB /˝2I2: Since A and B are disjoint and quasisymmetric, each non-zero block of D0 will have one of the following forms Ä Ä x x x x i i i i or i i i i ; jxi jxi jxi jxi Ä 11 : 1 where ` 2 S Multiplying D0 on the right by 2 Id ˝ 1 yields a matrix D1 of ˚ « order 2d with entries from 0; 1x1;:::;kxk whose non-zero 22 blocks have one of the forms Aixi or Bixi,where Ä Ä i 0 0i Ai D or Bi D ; (13) 0j j 0

andsuchthatD1D1 D D1 D1 D .AA C BB / ˝ I2: The Ai’s and Bi’s in (13)form another signed group, S0. Now matrices of the form Ä 0 ; ; ˝ I2 D 0 2 S form a signed subgroup of S0 which is isomorphic to S: Therefore, one can identify this signed subgroup with S itself and consider S0 as an extension of S: Replacing every 22 block of D1 which is one of the forms in (13) or zero with corresponding 0 ;0 0 : 0 i xi i 2 S or zero gives the required matrix D Note that we identify ˝ I2 2 S with 2 S: 0 Now if W S ! SPm Ä SPm is a faithful remrep of degree m, then it can be 0 0 0 veriﬁed directly that the map W S ! SP2m Ä SP2m which is uniquely deﬁned by Ä Ä Ä Ä 0 . /0 0 0 . / i ! i m ; i ! m i ; 0j 0m .j/ j 0 .j/0m is a faithful remrep of degree 2m,where0m denotes the zero matrix of order m: 116 E. Ghaderpour

Finally, since A and B are circulant, C consists of four circulant blocks, so D0 and D1 are block-circulant with block size 2 2I whence D is circulant and quasisymmetric. ut We now use Lemma 3.1 and follow similar techniques in [2, 13] to show the following Theorem.

Theorem 3.2. Suppose that B1;:::;Bn˚ are disjoint quasisymmetric« circulant matrices of order d with entries from 0; 1x1;:::;kxk ; where ` 2 SC; and the x`’s are variables .1 Ä ` Ä k/, such that Â Ã Xk 2 ; B1B1 CCBnBn D u`x` Id `D1 where the u`’s are positive integers. Then there is a quasisymmetric circulant n SOD dI u1;:::;uk for a signed group S that admits a faithful remrep of degree 2 . 0 Proof. SC has a faithful remrep W SC˝ ! SP2 Ä ˛SP2 of degree 2 uniquely 0 2 determined by .i/ D R,whereSP2 D RI R DI . Applying Lemma 3.1 to ; matrices B1 and B2 one obtains˚ a quasisymmetric« normal circulant matrix A1 of 0; .1/ ;:::;.1/ ; .1/ .1 ` / order d with entries from 1 x1 k xk where ` 2 S1 Ä Ä k 2 such that S1 SC is a signed group with a faithful remrep of degree 2 .Also, A1A1 D B1B1 C B2B2 : Since supp.A1/ is the union of supp.B1/ and supp.B2/; A1 is disjoint from B3;:::;Bn: Suppose that one has constructed˚ a circulant« quasisymmetric normal matrix Ar 0; .r/ ;:::;.r/ ; .r/ .1 ` / of order d with entries from 1 x1 k xk where ` 2 Sr Ä Ä k such 0 C1 that Sr Sr1 is a signed group with a faithful remrep r W Sr ! SP2rC1 Ä SP2r rC1 of degree 2 . Moreover, Ar is disjoint from BrC2;:::;Bn and

: ArAr D B1B1 CCBrC1BrC1

By˚ the assumption, B«rC2 is a quasisymmetric normal circulant matrix with entries from 0; 1x1;:::; kxk ; where ` 2 SC .1 Ä ` Ä k/: One can view the `’s as elements in Z Sr because we identiﬁed these elements as blocks ˙I2r ˝ R and .r/ ˙I2rC1 in the proof of Lemma 3.1 which commute with r ` ;1Ä ` Ä k: Therefore, by Lemma˚ 3.1, there is a quasisymmetric« normal circulant matrix ArC1 0; .rC1/ ;:::;.rC1/ ; .rC1/ .1 ` / with entries from 1 x1 k xk where ` 2 SrC1 Ä Ä k such rC2 that SrC1 Sr is a signed group with a faithful remrep of degree 2 .Also,

; ArC1ArC1 D ArAr C BrC2BrC2 D B1B1 CCBrC1BrC1 C BrC2BrC2 and by the same argument ArC1 is disjoint from BrC3;:::;Bn: Applying this procedure n 2 times, there is a quasisymmetric normal circulant matrix An1 of order d such that Signed Group Orthogonal Designs and Their Applications 117

Â Ã Xk 2 ; An1An1 D B1B1 CCBnBn D u`x` Id `D1 which is a circulant quasisymmetric SOD dI u1;:::;uk with the signed group n S D Sn1 Sn2 SC that admits a faithful remrep of degree 2 . ut Remark 3.1. The circulant matrices in Theorem 3.2 are taken on the abelian signed group SCI however, if the signed group is not abelian, the circulant matrices that obtain from Lemma 3.1 do not necessarily commute, and Theorem 3.2 may fail. As an example, if B1 D circ.j;0/and B2 D circ.0; k/; where j; k 2 SQ; then since jk Dkj; B1B2 ¤ B2B1: Therefore, Lemma 3.1 does not apply in this case.

Theorem 3.3. Suppose that B1;:::;Bn˚ are disjoint quasisymmetric« circulant matrices of order d with entries from 0; 1x1;:::;kxk ; where ` 2 SR; and the x`’s are variables .1 Ä ` Ä k/, such that Â Ã Xk 2 ; B1B1 CCBnBn D u`x` Id `D1 where the u`’s are positive integers. Then there is a circulant quasisymmetric SOD dI u1;:::;uk for a signed group S that admits a faithful remrep of degree 2n1.

Proof. Similar to the proof of Theorem 3.2, but in here since SR has the trivial remrep of degree 1, the ﬁnal signed group S will have a remrep of degree 2n1: ut Example 3.1. We explain how to use Theorem 3.3 to ﬁnd an SOD 12I 4;4;4 for a signed group S that admits a remrep of degree 8: Consider the following disjoint quasisymmetric circulant matrices of order 12: B1 D circa;0;0;0;0;0;a;0;0;0;0;0 , B2 D circ0; 0; 0; a;0;0;0;0;0;a;0;0, B3 D circ0; b; c;0;0;0;0;0;0;0;c; b , B4 D circ 0; 0; 0; 0; c; b;0;b; c;0;0;0 . 2 2 2 Thus, B1B1 C B2B2 C B3B3 C B4B4 D 4a C 4b C 4c I12: Apply Lemma 3.1 to B1 and B2 to get a quasisymmetric normal circulant matrix of order 12: A1 D circ 1a;0;0;ıa;0;0;1a;0;0;ıa;0;0 ; where ı is in the signed group of order 2: ˝ ˛ 2 S1 D 1; ıI ı D 1 which admits a remrep of degree 2 uniquely determined by 1 ! I2 and ı ! P: 2 Since B1 and B2 are complementary, it follows that A1A1 D 4a I12: 118 E. Ghaderpour

Applying Lemma 3.1 again to A1 and B3; there is a quasisymmetric normal circulant matrix of order 12: A1 D circ 1a;1b;2c;3a;0;0;1a;0;0;3a;2c; 1b ;

3 where 1;2;3 belong to the signed group of order 2 : D E 2 2 2 S2 D 1;2;3I 1 D2 D 3 D 1; ˛ˇ Dˇ˛I ˛; ˇ 2f1;2;3g ; with a remrep of degree 4 which is uniquely determined by

1 ! P ˝ I2;2 ! R ˝ I2;3 ! Q ˝ P:

Note that A2 is not an SOD because B1; B2,andB3 are not complementary. Finally, apply Lemma 3.1 to A2 and B4 to get a quasisymmetric normal circulant matrix of order 12: A3 D circ 1a;1b;2c;3a;4c; 5b;1a; 5b; 4c; 3a;2c; 1b ;

5 where j;1Ä j Ä 5 belong to the signed group of order 2 : D ; ; ; ; 2 2 2 2 2 1; S D 1 2 3 4 5I 1 D 2 D 3 D E4 D 5 D ˛ˇ Dˇ˛I ˛; ˇ 2f1;2;3;4;5g with a remrep of degree 8 which is uniquely determined by

1 ! Q˝P˝I2;2 ! Q˝R˝I2;3 ! Q˝Q˝P;4 ! P˝I2˝I2;5 ! R˝I2˝I2: So A3 is a quasisymmetric circulant SOD 12I 4;4;4 : By Theorem 3.1,thereisan OD 8 12I 8 4;8 4;8 4 :

Although Theorem 3.3 shows that the degree of remrep is 2 times less than the one in Theorem 3.2, we have more complex Golay pairs than real ones. Thus, from now on, we just consider the complex case, and we refer the reader to [7, chap. 6] for the results that obtain from the real case. For u a positive integer, denote by `c.u/ the least number of complex Golay numbers that add up to u,andlet`c.0/ D 0. Also, denote by `0c.u/ the least number of complex Golay numbers in two variables that add up to u. Indeed, `0c.2u/ Ä `c.u/: Note that Lemma 1.1 insures the existence of a complex Golay pair in two variables of length 2m if there exists a complex Golay pair of length m. In the following theorem, we show how to use complex Golay pair and complex Golay pairs in two variables to construct SODs. Signed Group Orthogonal Designs and Their Applications 119 1; v ;:::;v ; ; ;:::; ; Theorem 3.4. Let 1 q w1 w1 wt wPt be a sequenceP of positive inte- v 1 1 q v 2 t : gers such that i’s, Ä i Ä q, are disjoint. Let C ˇD1 ˇ C ıD1 wı D u Then a full circulant quasisymmetric SOD 4uI 4;4v1;:::;4vq;4w1;4w1;:::;4wt;4wt n existsP for some signed groupP S that admits a remrep of degree 2 ; where n Ä 2 2 q ` .v / 2 t ` . /: C ˇD1 c ˇ C ıD1 c wı Proof. For each 1 Ä ˇ Ä q, and each 1 Ä ˛ Ä `c.vˇ/,let AŒ˛; vˇI BŒ˛; vˇ be a complex Golay pair in one variable xˇ of length VŒ˛; vˇ : From the deﬁnition P ` .v / ˇ 1 ˇ ; `c.vˇ/ Œ˛; v v : Œ˛; ˇ Pof c ˇ , for eachP , Ä Ä q ˛D1 V ˇ D ˇ Let S WD ˛1 Œ ;v ˇ1 v : ı 1 ı 1 iD1 V i ˇ C jD1 j Also, for each , Ä Ä t, and each , Ä Ä 0 ` c.2wı/,let CŒ; wı I DŒ; wı be a complex Golay pair of length WŒ; wı in 0 two variables yı and zı. By the deﬁnition of ` c.2wı/, for each ı, 1 Ä ı Ä t; P 0 P P ` c.2wı/ Œ; 2 : 0Œ; ı 1 Œ ; 2 ı1 : ˇ D1 W wı D wı Let S WD iD1 W i wı C jD1 wj For each , 1 Ä ˇ Ä q, and each ˛, 1 Ä ˛ Ä `c.vˇ/, and forP each ı, 1 Ä ı ÄPt and each , 1 `0 .2 / 2 2 q ` .v / 2 t `0 .2 / Ä Ä c wı , the following are n D C ˇD1 c ˇ C ıD1 c wı circulant matrices of order 4u: M1 D circ x;0.2u1/; x;0.2u1/ ; M2 D circ 0.u/; x;0.2u1/; x;0.u1/ ; X˛ˇ D circ 0.SŒ˛;ˇC1/; AŒ˛; vˇ; 0.4u2SŒ˛C1;ˇ1/; BŒ˛; vˇ; 0.SŒ˛;ˇ/ ; Y˛ˇ D circ 0.2uSŒ˛C1;ˇ/; BŒ˛; vˇ; 0.2SŒ˛;ˇC1/; AŒ˛; vˇ ; 0.2uSŒ˛C1;ˇ1/ ; Zı D circ 0.vCS0Œ;ıC1/; CŒ; wı ; 0.4u2v2S0ŒC1;ı1/; DŒ; wı; 0.vCS0Œ;ı/ ; Tı D circ 0.2uvS0ŒC1;ı/;DŒ; wı ; 0.2S0Œ;ıC2vC1/; CŒ; wı ; 0.2uvS0 ŒC1;ı1/ :

It can be seen that the above circulant matrices are disjoint and quasisymmetric such that

` .v / 0 X2 Xq Xc ˇ Xt ` Xc.2wı/ MiMi C X˛ˇ X˛ˇ C Y˛ˇ Y˛ˇ C ZıZı C TıTı iD1 ˇD1 ˛D1 ıD1 D1 Â Ã Xq Xt 2 2 2 2 D 4 x C vˇxˇ C wıyı C wızı I4u: ˇD1 ıD1

Thus, by Theorem 3.2, there exists a full circulant quasisymmetric SOD 4uI 4;4v1;:::;4vq;4w1;4w1;:::;4wt;4wt for a signed group S which admits a remrep of degree 2n,where 120 E. Ghaderpour

Xq Xt Xq Xt 0 n D 2 C 2 `c.vˇ/ C 2 ` c.2wı/ Ä 2 C 2 `c.vˇ/ C 2 `c.wı/: ut ˇD1 ıD1 ˇD1 ıD1

Example 3.2. Consider the 4-tuple .1; v1;v2;v3/ D .1;5;7;17/. By Theorem 3.4, there is a circulant quasisymmetric SOD 430I 41; 45; 47; 4 17 ; which admits a remrep of degree 2n,wheren D 2C2`c.5/C2`c.7/C2`c.17/ D 2C2C4C4 D 12: By Theorem 3.1,thereisan OD 214 30I 214 1; 214 5; 214 7; 214 17 :

Example 3.3. Let .1; w1; w1; w2; w2; w3; w3; w4; w4/ D .1; 3; 3; 5; 5; 11; 11; 13; 13/. By Theorem 3.4, there is a circulant quasisymmetric SOD 4 65I 4 1; 4 3.2/;4 5.2/;4 11.2/;4 13.2/ ; which admits a remrep of degree 2n,wheren D 2 C 2`c.3/ C 2`c.5/ C 2`c.11/ C 2`c.13/ D 10: By Theorem 3.1,thereisan Á 12 12 12 12 12 12 OD 2 65I 2 1; 2 3.2/;2 5.2/;2 11.2/;2 13.2/ :

4 Bounds for the Asymptotic Existence Orthogonal Designs

In this section, we obtain some upper bounds for the degree of remrep in Theo- rem 3.4, and then we ﬁnd some upper bounds for the asymptotic existence of ODs. To get better upper bounds on the degree of remrep for any k-tuple u1; u2;:::;uk of positive integers, from now on, we assume that `c.u1/`c.u1 1/ is greater than or equal to `c.ui/ `c.ui 1/ for all 2 Ä i Ä k.Wealsodeﬁne log.0/ D 0, and in here the base of log is 2. Livinskyi [13, chap. 5], by a computer search, showed˘ that each positive integer u can be presented as sum of at most 3 log226 .u/ C 4 complex Golay numbers. Thus j 1 k 3 `c.u/ 3 .u/ 4 .u/ 4: Ä 26 log C Ä 26 log C (14) Theorem 4.1. Suppose that u1; u2;:::;uk is a k-tuple of positive integers and let u1 CCuk D u: Then a full circulant quasisymmetric SOD 4uI 4u1;4u2;:::;4uk n exists for some signed groupP S that admits a remrep of degree 2 ; where n Ä .3=13/ . 1/ .3=13/ k . / 8 2: log u1 C iD2 log ui C k C Signed Group Orthogonal Designs and Their Applications 121 Proof. Apply Theorem 3.4 to the .kC 1/-tuple 1; u1 1;u2;:::;uk .Sothereis 4 4 ;4 ;:::;4 a full circulant quasisymmetric SOD uI u1 u2 uk for some signedP group 2n; 2 2` . 1/ 2 k ` . /: S that admits a remrep of degree where n Ä C c u1 C iD2 c ui Use (14) to obtain the desired. ut Remark 4.1. For any givenk-tuple u1; u2;:::;uk of positive integers, one may write it as the .k C 1/-tuple 1; u1 1; u2;:::;uk , and then sort its elements to get the .k C 1/-tuple 1; v1;:::;vq; w1; w1;:::;wt; wt ; where vi’s are disjoint and then use Theorem 3.4 and (14) to obtain the following bound:

Xq Xt n Ä 2 C 2 `c.vi/ C 2 `c.wj/ iD1 jD1 3 Xq 3 Xt 2 .v / .w / 8.q t/; Ä C 13 log i C 13 log j C C iD1 jD1 where n is the exponent of the degree of remrep. By Theorems 3.1 and 4.1, we have the following asymptotic existence result. ; ;:::; Theorem 4.2. Suppose u1 u2 uk isP a k-tuple of positive integers. Then n k n n for each n N, there is an OD 2 u I 2 u1;:::;2 u ; where N Ä P jD1 j k .3=13/ . 1/ .3=13/ k . / 8 4: log u1 C iD2 log ui C k C Livinskyi [13, chap. 5] used complex Golay, Base, Normal and other sequences (see [5, 10–12]) to show that each positive integer u can be presented as sum of 1 s .u/ 5 Ä 10 log C (15) pairs .AkŒuI BkŒu/ for 1 Ä k Ä s such that AkŒu and ˚BkŒu have the same length for« each k, 1 Ä k Ä s, with elements from f˙1; ˙ig,and A1Œu; B1Œu;:::;AsŒu; BsŒu is a set of complex complementary sequences with weight 2u. In the following theorem, we use this set of complex complementary sequences. v ;v ;:::;v Theorem 4.3. Suppose 1 2P k is a k-tuple of positive integers. Then for n k n n each n N, thereis an OD 2 v I 2 v1;:::;2 v ; where N Ä .1=5/ log.v1 P jD1 j k 1/ .1=5/ k .v / 10 4: C iD2 log i C k C P v ;v ;:::;v k v v: Proof. Suppose 1 2 k is a k-tuple of positive integer. Let jD1 j D For simplicity, we assume that u1 ˚D v1 1 and ui D vi for 2 Ä i Ä k.« ˇ 1 ˇ Œ ; Œ ;:::; Œ ; Œ For each , Ä Ä k,let A1 uˇ B1 uˇ Asˇ uˇ Bsˇ uˇ be a set of complex complementary sequences with weight 2uˇ such that for each ˛, 1 Ä ˛ Ä sˇ, A˛Œuˇ and B˛Œuˇ have the same length, VŒ˛; uˇ.From(15), for each ˇ, 1 Ä ˇ Ä k, 122 E. Ghaderpour

1 sˇ uˇ 5: Ä 10 log C (16) Suppose that x and xˇ, 1 Ä ˇ Ä k are variables. Let M1 D circ x;0.2v1/; x;0.2v1/ and M2 D circ 0.v/; x;0.2v1/; x;0.v1/ : For each ˇ, 1 Ä ˇ Ä k, and each ˛, 1 Ä ˛ Ä sˇ,let Á X˛ˇ D circ 0.SŒ˛;ˇC1/; xˇA˛Œuˇ; 0.4v2SŒ˛C1;ˇ1/; xˇB˛Œuˇ; 0.SŒ˛;ˇ/ ; Á Y˛ˇ D circ 0.2vSŒ˛C1;ˇ/; xˇB˛Œuˇ; 0.2SŒ˛;ˇC1/; xˇA˛Œuˇ; 0.2vSŒ˛C1;ˇ1/ ; P P Œ1; 1 0 Œ ; a1 b Œ ; 1 where S D and S a b D jD1 iD1 V j ui ,for Ä b Ä k and 1

X2 Xk Xsˇ Xk Á 4 2 2 : MiMi C X˛ˇ X˛ˇ C Y˛ˇ Y˛ˇ D x C uˇxˇ I4v iD1 ˇD1 ˛D1 ˇD1 Thus, by Theorem 3.2, a full circulant quasisymmetric SOD 4vI 4;4u1;:::;4uk m existsP for a signed group S which admits a remrep of degree 2 ,wherem D 2 C 2 k : ˇD1 sˇ From Theorem 3.1 and the upper bounds for the sˇ’s, (16), there is an 2nv 2n;2n ;:::;2n ; 2nv 2nv ;:::;2nv ; OD I u1 uk andP so there is an OD I 1 k where .1=5/ .v 1/ .1=5/ k .v / 10 4 n Ä log 1 C iD2 log i C k C . ut

Acknowledgements The paper constitutes a part of the author’s Ph.D. thesis written under the direction of Professor Hadi Kharaghani at the University of Lethbridge. The author would like to thank Professor Hadi Kharaghani for introducing the problem and his very useful guidance toward solving the problem and also Professor Rob Craigen for his time and great help.

References

1. Craigen, R.: Constructions for Orthogonal Matrices. ProQuest LLC, Ann Arbor (1991) [Ph.D. Thesis, University of Waterloo, Canada] 2. Craigen, R.: Signed groups, sequences, and the asymptotic existence of Hadamard matrices. J. Comb. Theory Ser. A 71(2), 241–254 (1995) 3. Craigen, R., Holzmann, W.H., Kharaghani, H.: On the asymptotic existence of complex Hadamard matrices. J. Comb. Des. 5(5), 319–327 (1997) 4. Craigen, R., Holzmann, W., Kharaghani, H.: Complex Golay sequences: structure and applications. Discret. Math. 252(1–3), 73–89 (2002) 5. -Dokovic´, D.: On the base sequence conjecture. Discret. Math. 310(13–14), 1956–1964 (2010) 6. Geramita, A.V., Seberry, J.: Orthogonal Designs. Quadratic Forms and Hadamard Matrices. Lecture Notes in Pure and Applied Mathematics, vol. 45. Marcel Dekker Inc., New York (1979) Signed Group Orthogonal Designs and Their Applications 123

7. Ghaderpour, E.: Asymptotic Existence of Orthogonal Designs. ProQuest LLC, Ann Arbor (2013) [Ph.D. Thesis, University of Lethbridge, Canada] 8. Ghaderpour, E., Kharaghani, H.: The asymptotic existence of orthogonal designs. Australas. J. Comb. 58, 333–346 (2014) 9. Holzmann, W.H., Kharaghani, H.: On the amicability of orthogonal designs. J. Comb. Des. 17(3), 240–252 (2009) 10. Koukouvinos, C., Kounias, S., Sotirakoglou, K.: On base and Turyn sequences. Math. Comput. 55(192), 825–837 (1990) 11. Koukouvinos, C., Kounias, S., Seberry, J., Yang, C.H., Yang, J.: Multiplication of sequences with zero autocorrelation. Australas. J. Comb. 10, 5–15 (1994) 12. Kounias, S., Sotirakoglou, K.: Construction of orthogonal sequences. In: Proceedings of the 14th Greek Statistical Conference 192, pp. 229–236 (2001) 13. Livinskyi, I.: Asymptotic existence of Hadamard matrices. M.Sc. Thesis, University of Manitoba, Canada (2012) On Symmetric Designs and Binary 3-Frameproof Codes

Chuan Guo, Douglas R. Stinson, and Tran van Trung

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract In this paper, we study when the incidence matrix of a symmetric .v; k;/-BIBD is a 3-frameproof code. We show the existence of inﬁnite families of symmetric BIBDs that are 3-frameproof codes, as well as inﬁnite families of symmetric BIBDs that are not 3-frameproof codes.

Keywords Frameproof code • Hadamard design • Symmetric design

1 Introduction

Frameproof codes were introduced by Boneh and Shaw [2] as a method for digital rights control. Given a ﬁnite set Q and a positive integer N,letC Â QN be a ﬁnite set of length N codewords from the alphabet set Q. C is called an .N; n; q/ code if jCjDn and jQjDq. The elements of C are called codewords, with each codeword of the form x D .x1; x2;:::;xN/ where xi 2 Q for all i. C is called a w-frameproof code if no coalition of size at most w can construct a codeword not belonging to the coalition, and hence cannot frame the holder of the codeword. Formally, the coalition is a subset P Â C of w codewords. To construct a new codeword, for 1 Ä i Ä N, the coalition may select any available alphabet element ai from a codeword a 2 P and insert it into position i of the new codeword. The set of

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. C. Guo • D.R. Stinson () David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1 e-mail: [email protected]; [email protected] T. van Trung Institute for Experimental Mathematics, University of Duisburg-Essen, Essen, Germany e-mail: [email protected]

© Springer International Publishing Switzerland 2015 125 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_10 126 C. Guo et al.

N possible new codewords is desc.P/ Dfx 2 Q W xi 2fai W a 2 Pg;1 Ä i Ä Ng. Using this definition, a code C is w-frameproof if for any P Â C, jPjÄw,wehave desc.P/ \ C D P. The condition for C to be a frameproof code has been shown to be equivalent to a certain type of separating hash families. Let X; Y be finite sets and let H be a set of functions from X to Y. For pairwise disjoint subsets C1; C2;:::;Ct Â X, we say that h 2 H separates C1; C2;:::;Ct if h.C1/; h.C2/;:::;h.Ct/ Â Y are also pairwise disjoint. If some h 2 H separates C1; C2;:::;Ct for every choice of C1; C2;:::;Ct with jCijDwi for fixed integers wi, i D 1;:::;N, we say that H is an .NI n; q; fw1; w2;:::;wtg/-separating hash family (denoted SHF.NI n; q; fw1; w2;:::;wtg/), where N DjH j, n DjXj, q DjYj. .N; n; q/ w- frameproof codes are equivalent to SHF.NI n; q; f1; wg/ [14], and it is usually easier to study frameproof codes in terms of the equivalent separating hash family. Given an SHF.NI n; q; fw1; w2;:::;wtg/, we may represent it as an N n matrix with entries 1;:::;q. The rows are indexed by H and the columns are indexed by X.Thevalueofh.x/ is the entry in the matrix at row h and column x. An N n matrix A with entries 1;:::;q is the representation matrix of an SHF.NI n; q; fw1; w2;:::;wtg/ if and only if for every pairwise disjoint set of columns C1; C2;:::;Ct with jCijDwi, i D 1;:::;N, there exists a row h such that fA.h; x/ W x 2 Cig\fA.h; x/ W x 2 CjgD;for every i ¤ j. It is easy to see that a permutation matrix of order N always gives an SHF.NI N;2;f1; wg/ for any w.In[9], it was shown that for w 3,ifN Ä 3w then SHF.NI n;2; f1; wg/ exists if and only if n Ä N and permutation matrices are the only examples of SHF.NI N;2;f1; wg/ up to exchanging 0s and 1s in each row independently. One interesting question is finding examples of SHF.NI N;2;f1; wg/ with N >3w that are not permutation matrices. In this paper, we will explore the option of using symmetric BIBDs to provide such examples. The rest of the paper is organized as follows. In Sect. 2, we mention some previous results. Section 3 contains our main results. First we give a parametric and structural characterization of symmetric BIBDs that are 3-frameproof codes. In Sect. 3.1, we study the case of Hadamard designs in detail, and Sect. 3.2 addresses designs with k D 3. We consider “small” designs in Sect. 3.3 and we construct a table of existence and nonexistence results. Finally, Sect. 4 contains a couple of additional results and some open problems.

2 Previous Results

Connections between balanced incomplete block design (BIBDs) and SHFs have previously been discovered. Type f1; wg SHFs can be derived from t-designs—a more general form of combinatorial designs. A t-.v; k;/ design is a set system .X; B/ where X is a ﬁnite set and B is a set of subsets of X satisfying On Symmetric Designs and Binary 3-Frameproof Codes 127

(i) jXjDv, (ii) jBjDk for every B 2 B,and (iii) for every Y Â X, jYjDt,thereexistB1; B2;:::;B 2 B distinct such that Y Â Bi for i D 1;:::;. The elements of X are called points and the elements of B are called blocks. In the special case that t D 2, we call the design a .v; k;/-BIBD. The number of blocks of the design is b DjBj, and is related to the parameters of the design by bk D vr, .v1/ v where r D k1 is the number of blocks each point is contained in. If D b, the design is called a symmetric BIBD (denoted SBIBD). One family of symmetric BIBDs is the projective planes with v D q2 C q C 1, k D q C 1 and D 1,whereq is a prime power. It is often useful to represent a t-design using a binary matrix. Given a t-.v; k;/ design .X; B/, its point-block incidence matrix is the vb binary matrix A with rows indexed by X and columns indexed by B, with A.x; B/ D 1 if and only if x 2 B.The block-point incidence matrix is the transpose of the point-block incidence matrix. The following theorem gives a construction for SHFs from t-designs. Theorem 2.1 ([13]). Let .X; B/ be a t-.v; k;1/ design. The point-block incidence . ; B/ .v ;2; 1; / v = k matrix of X is an SHF I b f wg where b D t t is the number of k1 blocks and w Dbt1 c. For t D 2, this result shows that all members of the projective plane family of designs is an SHF.vI v; 2; f1; wg/ for w D k 1.

3 Binary Frameproof Codes

In this section, we give a characterization of SHF.vI v; 2; f1; 3g/ for all symmetric .v; k;/-BIBDs. Theorem 3.1. Let .X; B/ be a symmetric .v; k;/-BIBD and let A be its block- point incidence matrix. If k 3 C 1 or if k is odd, then A is an SHF.vI v; 2; f1; 3g/. Proof. Suppose that A is not an SHF.vI v; 2; f1; 3g/, then there exists some column set pair .fxg; fu; v; wg/ that cannot be separated. For each Z Âfu; v; wg, partition B into subsets AZ where AZ DfB 2 B W B \fu; v; wgDZg,andletaZ DjAZj. We obtain the following set of equations from .X; B/ being a symmetric .v; k;/- BIBD:

a; C au C av C aw C auv C avw C auw C auvw D v (1)

au C auv C auw C auvw D k (2)

av C auv C avw C auvw D k (3) 128 C. Guo et al.

aw C auw C avw C auvw D k (4)

auv C auvw D (5)

avw C auvw D (6)

auw C auvw D (7)

Letting ˛ D auvw, we get that

auv D avw D auw D ˛

au D av D aw D k 2. ˛/ ˛ D k C ˛ 2:

Next, deﬁne BZ DfB 2 AZ W x 2 Bg and let bZ DjBZ j. We obtain another set of equations:

b; C bu C bv C bw C buv C bvw C buw C buvw D k (8)

bu C buv C buw C buvw D (9)

bv C buv C bvw C buvw D (10)

bw C buw C bvw C buvw D (11)

Note that for every Z,weget0 Ä bZ Ä aZ. It is clear that the column set pair .fxg; fu; v; wg/ cannot be separated if and only if b; D 0 and buvw D ˛. Thus Eqs. (8)–(11) simplify to

bu C bv C bw C buv C bvw C buw D k ˛ (12)

bu C bv C bw C 2.buv C bvw C buw/ D 3. ˛/ (13)

Subtracting (12) from (13)gives

buv C bvw C buw D 3 k 2˛: (14)

Since buv C bvw C buw 0,(14) implies that

0 Ä 3 k 2˛: (15)

Now, since ˛ 0, we see from (15)thatk Ä 3. Therefore A is an SHF.vI v; 2; f1; 3g/ if k 3 C 1. Next, we multiply (12) by 2 and subtract (13), giving

bu C bv C bw D ˛ C 2k 3: (16) On Symmetric Designs and Binary 3-Frameproof Codes 129

Then we have

3.k C ˛ 2/ D au C av C aw bu C bv C bw D ˛ C 2k 3: (17)

Therefore, from (17), we have

3 k 2˛ Ä 0: (18)

Now, (15)and(18) together show that 3k D 2˛. This implies that 3k is even, and therefore k is also even. Therefore A is an SHF.vI v; 2; f1; 3g/ if k is odd. ut Corollary 3.1. Let .X; B/ be a symmetric .v; k;/-BIBD and let A be its block- point incidence matrix. If k Ä 3 and k is even, then A is an SHF.vI v; 2; f1; 3g/ if and only if there does not exist four points u; v; w; x such that ˛ 3k ; ; ; 1. D 2 blocks contain all four points u v w x, 2. no block contains exactly one or three points from fu; v; w; xg, and 3. for any subset of two points from fu; v; w; xg, there are exactly ˛ blocks that contain these two points. Proof. It is clear that A is not an SHF.vI v; 2; f1; 3g/ if the speciﬁed four-point substructure exists. So we just need to prove the converse, namely, that the four- point substructure exists if A is not an SHF.vI v; 2; f1; 3g/. We use the same notation as in the proof of Theorem 3.1. The proof of that theorem established that ˛ D .3 k/=2. For each T Âfu; v; w; xg, we will compute cT , which denotes the number of blocks B such that B \fu; v; w; xgDT. First, we note two relevant facts:

• The inequality in (17) must be an inequality, so bu D au, bv D av and bw D aw. Now au D av D aw D k C˛ 2 D ˛, so we obtain bu D bv D bw D ˛. •From(14), we see that buv C bvw C buw D 0,sobuv D bvw D buw D 0.

It is now straightforward to compute the values cT using these facts. This is done in Table 1. ut

Table 1 Block intersections with fu; v; w; xg

T cT T cT

fxg b; D 0 fug au bu D 0

fvg av bv D 0 fwg av bv D 0

fu; xg bu D ˛ fu; vg auv buv D auv D ˛

fv; xg bv D ˛ fu; wg auw buw D auw D ˛

fw; xg bw D ˛ fv; wg avw bvw D avw D ˛

fu; v; xg buv D 0 fu; w; xg buw D 0

fv; w; xg bvw D 0 fu; v; wg auvw buvw D ˛ ˛ D 0

fu; v; w; xg buvw D ˛ 130 C. Guo et al.

3.1 Hadamard Designs

The following is an immediate corollary of Theorem 3.1. This result is in fact equivalent to a result of Kimura [11, Proposition 2.1]. Corollary 3.2. When n >1is odd, the incidence matrix of a .4n1; 2n1; n1/- SBIBD is an SHF.4n 1I 4n 1; 2; f1; 3g/. There is a useful classiﬁcation of Hadamard matrices in terms of substructures involving four columns; see, for example, [10]. The notion of a type of a Hadamard matrix is deﬁned in [10] as follows. Let H be a Hadamard matrix of order 4n.For any non-negative integer m,letjm denote the all 1’s column vector of length m.By permuting and/or and negating rows and columns, any four columns of H may be transformed uniquely to the following form:

ja ja ja ja jb jb jb jb jb jb jb jb ja ja ja ja jb jb jb jb ja ja ja ja ja ja ja ja jb jb jb jb where a C b D n and 0 Ä b Äbn=2c. A set of four columns which is transformed to the above form is said to be of type b. Any permutation and negation of rows and/or columns leaves the type unchanged. A Hadamard matrix is of type b (0 Ä b Äbn=2c) if it has a set of four columns of type b and no set of four columns of type less than b. If a Hadamard matrix has a first row and first column consisting entirely of entries equal to 1, then we say that the matrix is standardized. Any Hadamard matrix can be transformed into a standardized Hadamard matrix by multiplying certain rows and columns by 1. Lemma 3.1. Suppose we construct an incidence matrix of a .4n 1; 2n 1; n 1/- SBIBD from a standardized Hadamard matrix of order 4n >4by deleting the first row and column and replacing all occurrences of 1’s by 0’s. Then this incidence matrix is a 3-frameproof code if and only if the Hadamard matrix is not of type 0. Proof. First, suppose that the Hadamard is of type 0. Then it is obvious in the incidence matrix of the associated design that the first of the four given columns cannot be separated from the other three given columns. Conversely, suppose that we have an incidence matrix A (of a .4n1; 2n1; n 1/-SBIBD) that is not a 3-frameproof code. From Corollary 3.2, n must be even for this to occur. By permuting columns of A, we can assume that column 1 cannot be On Symmetric Designs and Binary 3-Frameproof Codes 131 separated from columns 2, 3, and 4. Now we apply Corollary 3.1. Looking at the first four columns of A, there must be n=2 1 occurrences of 1111 and n=2 occurrences of each of the other seven patterns containing an even number of 1’s. When we convert A to a Hadamard matrix H of order 4n, we change all 0’s to 1’s and we add an additional row of 1’s. Now we multiply all rows of H that corresponded to patterns 0000, 0011, 0101,and0110 in A by 1. We then see that these four columns in H are of type 0. ut Remark. Kimura’s result that was mentioned above is in fact a proof that a Hadamard matrix of order congruent to 4 modulo 8 is not of type 0. A classification, according to type, of (inequivalent) Hadamard matrices of small orders is given in [10]. Table 2 is from [10]:

Table 2 Number of inequivalent Hadamard matrices of different types Order 4 8 12 16 20 24 28 0 11050580 Type 1 001031486 2 000001 1

We now give a family of Hadamard BIBDs that contain the forbidden substructure from Corollary 3.1. Hence, these designs are not f1; 3g-SHFs.

Theorem 3.2. For n 4,letHn be a standardized Hadamard matrix of order n. Let Â Ã H H H D n n Hn Hn and let A be the .2n 1/ .2n 1/ submatrix of H by removing the ﬁrst column and ﬁrst row and replacing all 1’s by 0’s. Then A is the incidence matrix of a .2 1; 1; n2 / .2 1 2 1; 2; 1; 3 / symmetric n n 2 -design which is not an SHF n I n f g .

Proof. A is a Hadamard design by construction. Let n D 4m, m 1.SinceHn is a standard Hadamard matrix of order 4m, deleting the first column gives a 2- .2; 4m 1; m/ orthogonal array. Hence columns 2 and 3 of Hn contain each of the pairs .0; 0/, .0; 1/, .1; 0/, .1; 1/ m times. Thus columns 2;3;4m C 2;4m C 3 of H contain each of the quadruples .0;0;0;0/, .0;1;0;1/, .1;0;1;0/, .1;1;1;1/ m times in rows 1;:::;4m of H. Similarly, columns 2;3;4m C 2;4m C 3 of H contain each of the quadruples .0;0;1;1/, .0;1;1;0/, .1;0;0;1/, .1;1;0;0/m times in rows 4m C 1;:::;8m of H. Recall that the first column of H is deleted to form A. Since the first row of H consists of only 1’s, we have that columns 1; 2; 4m C 1; 4m C 2 of A contain each of the quadruples .0;0;0;0/, .0;1;0;1/, .1;0;1;0/, .0;0;1;1/, .0;1;1;0/, .1;0;0;1/, .1;1;0;0/ m times and contains .1;1;1;1/ m 1 times. Together, the eight 132 C. Guo et al. quadruples occupy all 8m1 rows of A. In particular, columns 1; 2; 4mC1; 4mC2 of A do not contain the quadruple .1;0;0;0/and .0;1;1;1/,so.f1g; f2;4m C 1; 4m C 2g/ cannot be separated by A. ut The quadratic residue difference sets (also known as Paley difference sets)give rise to Hadamard designs. For a prime power q Á 3 mod 4, the set of quadratic residues in Fq, when developed through Fq, yields a .q;.q1/=2; .q3/=4/-SBIBD. When q >11is prime, we will show that the incidence matrices of these designs are f1; 3g-SHFs. The proof is similar to the main theorem in [8]; it is based on a character-theoretic bound proven by Burgess [3]. Theorem 3.3. For all primes q Á 3 mod 4,q>11,thereisa.q;.q 1/=2; .q 3/=4/-SBIBD whose block-point incidence matrix is a f1; 3g-SHF. Z 1; 1 .0/ 0 Proof. Let W q !f g be the quadratic character. Define D and let a1; a2; a3; a4 2 Zq be distinct. Define X S D .x a1/.x a2/.x a3/.x a4/: (19)

x2Zq p By [3, Lemma 1], it immediately follows that S Ä 2 qC1. For any integer q >11, p it is easy to see that 2 q C 1 1;024, it is noted in Colbourn and Kéri [5] that Paley difference sets yield covering arrays of strength four, which immediately implies that they are f1; 3g-SHFs. This follows from a similar character-theoretic argument. Theorem 3.4. There is a .39; 19; 9/-SBIBD whose incidence matrix is a f1; 3g-SHF. Proof. The website [12] includes 22 (known to date) skew Hadamard matrices of order 40. We derived Hadamard designs (i.e., .39; 19; 9/-SBIBDS) from all of them by standardizing with respect to a given row and column and then deleting the given row and column. Then we checked the resulting .39; 19; 9/-SBIBDs by computer to see if they are f1; 3g-SHF. It turned out that eight of these matrices, namely numbers 1, 5, 7, 10, 11, 13, 17, and 20, give rise to .39; 19; 9/-SBIBDs which are f1; 3g-SHF. Moreover, the transposes of the incidence matrices of these 22 .39; 19; 9/-SBIBDs give rise to eight additional .39; 19; 9/-SBIBDs which are f1; 3g-SHF, namely numbers 2, 3, 8, 12, 14, 16, 18, and 21. It did not matter which row/column we chose for the standardization process. ut On Symmetric Designs and Binary 3-Frameproof Codes 133

3.2 The Case k D 3

The case k D 3 is especially interesting because this corresponds to ˛ D 0 in Theorem 3.1. In this situation, the four-point substructure is an oval,usingthe terminology of Assmus and van Lint [1] (the paper [1] is a general study of ovals in symmetric BIBDs). Specializing Corollary 3.1 to this case, we obtain the following. Corollary 3.3. Let .X; A / be a symmetric .v; k;/-BIBD with k D 3.Then .X; A / is not an SHF.vI v; 2; f1; 3g/ if and only if .X; A / contains an oval (of cardinality 4). We next present some examples to show how Corollary 3.3 can be used to determine if a specific parameter set gives rise to f1; 3g-SHFs. Example 3.1. There is a unique .7;3;1/-SBIBD up to isomorphism. As is observed in [1], the complement of any block is an oval. Therefore the .7;3;1/-SBIBD is not a f1; 3g-SHF. Example 3.2. There are precisely three non-isomorphic .16; 6; 2/-SBIBDs. It is observed in [1] that all three of these designs contain ovals. Therefore, no .16; 6; 2/- SBIBD is a f1; 3g-SHF. Example 3.3. It is observed in [1] that there is a .25; 9; 3/-SBIBD that contains an oval. Therefore this SBIBD is not a f1; 3g-SHF. In fact, Denniston later showed in [7] that all 78 non-isomorphic .25; 9; 3/-SBIBDs contain an oval, so there are no .25; 9; 3/-SBIBDs whose incidence matrices are f1; 3g-SHFs. Finally, we present an infinite family of symmetric BIBDs with k D 3 whose incidence matrices are not f1; 3g-SHFs. Theorem 3.5. For all integers h 2,thereisa.3hC1 2;3h;3h1/-SBIBD whose incidence matrix is not a f1; 3g-SHF. Proof. It is shown by Tran in [15] that the Mitchell–Rajkundlia designs with the above parameters all contain ovals. (Actually, Tran shows that the Mitchell– Rajkundlia designs constructed from the Desarguesian affine planes of order 2n all contain maximal 2m-arcs for 1 Ä m Ä n. For the specific Mitchell–Rajkundlia designs with the indicated parameters, we have m D 1, and the maximal 2-arcs are in fact ovals.) ut

3.3 Small Symmetric BIBDs

Table 3 lists parameters for ‘small’ symmetric BIBDs and constructions that give rise to f1; 3g-SHFs (or not). The case of D 1 for k 4 is characterized by Theorem 2.1 and so these parameters are omitted from the table. 134 C. Guo et al.

Table 3 Small symmetric BIBDs and f1; 3g-SHF v k f1; 3g-SHF not f1; 3g-SHF Comment 7 3 1 None All Example 3.1 11 5 2 All None T2 16 6 2 None All Example 3.2 15 7 3 None All Table 2 37 9 2 All None T1 25 9 3 None All Example 3.3 19 9 4 All None T2 31 10 3 All None T1 56 11 2 All None T1 23 11 5 QR.23/ H 45 12 3 All None T1 79 13 2 All None T1 40 13 4 All None T1 27 13 6 All None T2 71 15 3 All None T1 36 15 6 All None T2 31 15 7 QR.31/ H 61 16 4 All None T1 49 16 5 All None T1 41 16 6 [4, §II.6.9] ? Computer veriﬁed 69 17 4 All None T1 35 17 8 All None T2 39 19 9 Theorem 3.4 H 96 20 4 All None T1 85 21 5 All None T1 71 21 6 All None T1 43 21 10 All None T2 78 22 6 All None T1 47 23 11 QR.47/ H 70 24 8 [4, §II.6.9] ? Computer veriﬁed 121 25 5 All None T1 101 25 6 All None T1 61 25 10 All None T2 51 25 12 All None T2

Legend Description T1 Guaranteed to be {1,3}-SHFs by Theorem 3.1 from k 3 C 1 T2 Guaranteed to be {1,3}-SHFs by Theorem 3.1 from k odd H Construction from Theorem 3.2 QR.q/ Quadratic residue difference set (Theorem 3.3) On Symmetric Designs and Binary 3-Frameproof Codes 135

4 Additional Results and Comments

We have a simple result which shows that certain symmetric BIBDs are f1; wg-SHF. Theorem 4.1. Suppose there exists a symmetric .v; k;/-BIBD where k > w. Then the block-point incidence matrix of this SBIBD is a f1; wg-SHF. Proof. Let A be the block-point incidence matrix of the hypothesized design. Let i be one column of A and let j1;:::;jw be w additional columns of A .For1 Ä ` Ä w, deﬁne

R` Dfr W A.r; i/ D A.r; j`/ D 1g:

Clearly jR`jD for all `,so ˇ ˇ ˇ ˇ ˇ[w ˇ ˇ ˇ : ˇ R`ˇ Ä w `D1

There are k rows of A having a 1 in column i.Sincek > w, there exists at least one row of A having a 1 in column i and 0’s in columns j1;:::;jw. ut Remark. In the case w D 3, Theorem 4.1 provides a simple proof of the ﬁrst part of Theorem 3.1.

Deﬁne Fw to be the set of all parameter triples .v; k;/ such that there exists a symmetric .v; k;/-BIBD whose incidence matrix is a f1; wg-SHF, and deﬁne F w to be the set of all parameter triples .v; k;/ such that there exists a symmetric .v; k;/-BIBD whose incidence matrix is not a f1; wg-SHF. A parameter triple .v; k;/will be called a Hadamard triple if it has the form .4t C 3; 2t C 1; t/ for a positive integer t,andanon-Hadamard triple otherwise. There are several parameter triples in Table 3 that are in F3 \ F 3.However, all of these examples are Hadamard triples. We now provide an example of a non- Hadamard triple in F3 \ F 3, namely .64; 28; 12/. Theorem 4.2. There exists a .64; 28; 12/-SBIBD whose incidence matrix is a f1; 3g-SHF,aswellasa.64; 28; 12/-SBIBD whose incidence matrix is not a f1; 3g- SHF.

Proof. We have veriﬁed by computer that the incidence matrix of the design D1 in [6, p. 113] is a f1; 3g-SHF. Furthermore, the incidence matrix of the design constructed from the difference set in Z4 Z16 (see [4, p. 428]) is not a f1; 3g- SHF. ut We close the paper by mentioning three open problems.

1. From the results proven in this paper, we know that F3 contains an inﬁnite number of Hadamard triples, and F 3 also contains an inﬁnite number of 136 C. Guo et al.

Hadamard triples. We conjecture that F3 \ F 3 also contains an inﬁnite number of Hadamard triples. 2. Does F3 contain an inﬁnite number of non-Hadamard triples which do not satisfy one of the hypotheses of Theorem 3.1? 3. It was proven in [9]thatanSHF.vI v; 2; f1; 3g/ is “equivalent” to a v v permutation matrix if v Ä 9. We have shown in this paper that an SHF.11I 11; 2; f1; 3g/ can be obtained from the incidence matrix of an .11; 5; 2/-SBIBD. The case v D 10 is open: it is not known if there exists an SHF.10I 10; 2; f1; 3g/ that is not equivalent to a permutation matrix.

Acknowledgements Thanks to Charlie Colbourn, Hadi Kharaghani, William Orrick, and Behruz Tayfeh-Rezaie for helpful comments and for making us aware of some relevant papers. D. Stinson’s research is supported by NSERC discovery grant 203114-11.

References

1. Assmus, Jr., E.F., van Lint, J.H.: Ovals in projective designs. J. Comb. Theory Ser. A 27, 307–324 (1979) 2. Boneh, D., Shaw, J.: Collusion-secure fingerprinting for digital data. IEEE Trans. Inf. Theory 44, 1897–1905 (1998) 3. Burgess, D.A.: On character sums and primitive roots. Proc. Lond. Math. Soc. 12, 179–192 (1962) 4. Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs, 2nd edn. Chapman & Hall/CRC, Boca Raton (2006) 5. Colbourn, C.J., Kéri, G.: Binary covering arrays and existentially closed graphs. Lect. Notes Comput. Sci. 5557, 22–33 (2009) (IWCC 2009 Proceedings) 6. Crnkovic,´ D., Pavcevi˘ c,´ M.-O.: Some new symmetric designs with parameters .64; 28; 12/. Discret. Math. 237, 109–118 (2001) 7. Denniston, R.H.F.: Enumeration of symmetric designs (25,9,3). Ann. Discret. Math. 15, 111–127 (1982) 8. Graham, R.L., Spencer, J.H.: A constructive solution to a tournament problem. Can. Math. Bull. 14, 45–47 (1971) 9. Guo, C., Stinson, D.R., van Trung, T.: On tight bounds for binary frameproof codes. Des. Codes Crypt. (to appear) 10. Kharaghani, H., Tayfeh-Rezaie, B.: On the classification of Hadamard matrices of order 32. J. Comb. Des. 18, 328–336 (2010) 11. Kimura, H.: Classification of Hadamard matrices of order 28. Discret. Math. 133, 171–180 (1994) 12. Koukouvinos, C.: www.math.ntua.gr/people/(ckoukouv2)/hadamard.htm. 13. Stinson, D.R., Wei, R.: Combinatorial properties and constructions of traceability schemes and frameproof codes. SIAM J. Discret. Math. 11, 41–53 (1998) 14. Stinson, D.R., van Trung, T., Wei, R.: Secure frameproof codes, key distribution patterns, group testing algorithms and related structures. J. Stat. Plann. Inference 86, 595–617 (2000) 15. van Trung, T.: Maximal arcs and related designs. J. Comb. Theory Ser. A 57, 294–301 (1991) An Algorithm for Constructing Hjelmslev Planes

Joanne L. Hall and Asha Rao

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract Projective Hjelmslev planes and affine Hjelmslev planes are generalisations of projective planes and affine planes. We present an algorithm for constructing projective Hjelmslev planes and affine Hjelmslev planes that uses projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes and all 2-uniform affine Hjelmslev planes can be constructed in this way. As a corollary it is shown that all 2-uniform affine Hjelmslev planes are sub-geometries of 2-uniform projective Hjelmslev planes.

Keywords Projective Hjemslev planes • Afﬁne Hjelmslev planes • 2-uniform • algorithm

1 Introduction

In 1916, Hjelmslev introduced the concept of a projective Hjelmslev geometry as a “geometry of reality” [11], introducing the intriguing concept of point and line neighbourhoods, a property that varied the restriction that two points lie on exactly one line. However it was the 1950s before Klingenberg ﬁrst formally deﬁned Hjelmslev planes [18]. More work describing these geometries was done in the 1960s and 1970s by Drake and others [5, 7, 8].

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. J.L. Hall School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4000, Australia e-mail: [email protected] A. Rao () School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, VIC 3001, Australia e-mail: [email protected]

© Springer International Publishing Switzerland 2015 137 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_11 138 J.L. Hall and A. Rao

Recently there has been renewed interest in these structures with Honold and Landjev [13] showing a connection with linear codes and Saniga and Planat [19] conjecturing that there may be connections to mutually unbiased bases. A difficulty turns out to be explicit constructions and concrete examples, which would help further investigations. Some Hjelmslev planes can be constructed using chain rings [21], those that have been the most thoroughly investigated have been the Galois ring Hjelmslev geometries [13, 15, 16], though there has been some more recent work using other chain rings [12, 14]. Note that just as there are affine planes and projective planes which cannot be constructed via a Galois field, similarly there are Hjelmslev planes which cannot be constructed using a chain ring. Thus, having a general algorithm which generates Hjelmslev planes regardless of their algebraic structure would be useful. In this paper we give such an algorithm for constructing projective Hjelmslev planes, using a projective plane, an affine plane and an orthogonal array. This algorithm is easily implemented using any programming language and generates a Hjelmslev plane for use in applications. Indeed it has been implemented using Python and in the conclusion, we provide a link to a visualisation. There are open online lookup tables (e.g. [2, 20]) for the structures needed for this construction, or some may be constructed using a Galois field [4, §VII.2]. Drake and Shult [9, Prop 2.4] show that all Hjelmslev planes can be constructed using a projective plane and semi-nets with zings, however there is no library of semi-nets with zings. While our Algorithm 4.1 for constructing Projective Hjelmslev planes is implied by the proof of [10, Thm 1], we present it here in algorithmic language. In addition the Algorithm 5.1 for constructing affine Hjelmslev planes is new. All affine planes are sub-geometries of projective planes, however there exist affine Hjelmslev planes which are not sub-geometries of projective Hjelmslev planes [1]. A result of the presented algorithms is a new proof showing that all 2-uniform affine Hjelmslev planes are sub-geometries of projective Hjelmslev planes. While the major drawback of this algorithm is the large number (factorial in the order of the projective plane) of isomorphic Hjelmslev planes which are generated via different paths, the algorithm is still useful in providing examples and allowing for visualisation of these beautiful, but intricate structures. The authors have been as yet unable to come up with a way to address the problem of Hjelmslev isomorphisms. A 1989 paper of Hanssens and Van Maldeghm [10]givessome results regarding isomorphisms of 3- and higher uniform Hjelmslev planes, but the same cannot be said for 2-uniform Hjelmslev planes which seems to be a more difficult problem. The paper is organised in the following manner. Section 2 gives definitions and facts about projective Hjelmslev planes, while Sect. 3 describes the mathematical objects required for the construction. Sections 4 and 5 detail algorithms for constructing 2-uniform projective Hjelmslev planes and 2-uniform affine Hjelmslev planes. Section 6 discusses some properties of the algorithms and concludes with some future directions. An Algorithm for Constructing Hjelmslev Planes 139

2 Deﬁnitions

Hjelmslev planes are generalisations of projective and affine planes, see, for example, [4, 6]. We write P 2 h, to indicate that the point P is incident with line h. A line may also be represented as the set of points incident with it. In Hjelmslev geometry, lines (or points) may have the relationship of being neighbours, denoted g h (or P Q). Definition 2.1 ([8]). A projective Hjelmslev plane, H , is an incidence structure such that 1. any two points are incident with at least one line. 2. any two lines intersect in at least one point. 3. any two lines g, h, that intersect at more than one point are neighbours, denoted g h. 4. any two points P and Q that are incident with more than one line are neighbours, denoted P Q. 5. there exists an epimorphism from H to an ordinary projective plane P such that for any points P; Q and lines g; h of H : (a) .P/ D .Q/ ” P Q, (b) .g/ D .h/ ” g h. The neighbour property is an equivalence relation, so the set of lines of H is partitioned into line-neighbourhoods and the set of points of H is partitioned into point-neighbourhoods. A projective Hjelmslev plane is denoted by .t; r/PH-plane where t is the number of neighbouring points on each line, and r is the order of the projective plane associated by the epimorphism . Furthermore it is known that r D s=t where t C s is the number of points on each line, each line-neighbourhood has t2 lines and each point-neighbourhood has t2 points [8, 17]. Just as in projective planes, points and lines are dual. A .1; r/PH-plane is a projective plane of order r. This notation should not be confused with PH.R/, the projective Hjelmslev plane over the ring R,orPH.n; pr/ where R D GR.pn; r/ [13, 19]. Definition 2.2 ([8]). An affine Hjelmslev plane H is an incidence structure such that 1. any two points are incident with at least one line. 2. any two lines g, h that intersect at more than one point are neighbours, denoted g h. 3. There exists an epimorphism from H to an ordinary affine plane A such that for any points P; Q and lines g; h of H : (a) .P/ D .Q/ ” P Q, (b) .g/ D .h/ ” g h, (c) jg \ hjD0 ) .g/ k .h/. 140 J.L. Hall and A. Rao

An affine Hjelmslev plane may be denoted .t; r/AH-plane, where t is the number of neighbouring points on each line, and r D s=t where s is the number of points on each line. An affine Hjelmslev plane may have parallel lines which are neighbours as condition 3c of Definition 2.2 is a one way implication. Hjelmslev planes are mentioned in some books on finite geometry [6, §7.2], but not in the more standard works on design theory e.g. [3, 4]. Hjelmslev planes have a rich structure with several interesting substructures. The following function allows interrogation of the structure of each point neighbourhood. Definition 2.3 ([7, Defi 2.3]). Let P be a point of a Hjelmslev plane H . The point- neighbourhood restriction, PN ,isdefinedasfollows: 1. the points of PN are the points Q of H such that Q P. 2. the lines of PN are the restrictions of lines g of H to the points in PN : gP D g \ PN . Definition 2.4 ([7, Defi 2.4]). A 1-uniform projective (affine) Hjelmslev plane H is an ordinary projective (affine) plane. A projective (affine) Hjelmslev plane is n- uniform if 1. for every point P 2 H , PN is an .n 1/ uniform affine Hjelmslev plane. 2. for each point P of H , every line of PN is the restriction of the same number of lines of H . In a 2-uniform projective Hjelmslev plane every point-neighbourhood restriction is an ordinary affine plane. It is known that a projective Hjelmslev plane is 2-uniform if and only if it is a .t; t/PH-plane, similarly an affine Hjelmslev plane is 2-uniform if and only if it is a .t; t/AH-plane [17]. All .t; t/PH-planes that can be generated by rings have been catalogued [13]. However there are many more Hjelmslev planes that cannot be generated by rings.

3 The Objects Required for the Construction

We present an algorithm for generating a 2-uniform projective Hjelmslev plane. This algorithm is inspired by a construction by Drake and Shult [9,Prop2.4]anda construction by Hanssens and VanMaldeghem [10]. The construction of Drake and Schult uses a projective plane and a semi-net with zings; semi-nets have not been well catalogued, making Drake and Shult’s construction difﬁcult to implement. The construction of Hanssens and Van Maldeghem is not given in algorithmic language. The algorithms given in Sects. 4 and 5 are restricted to .t; t/PH-planes and .t; t/AH- planes, but use objects that are well studied, and catalogued. The algorithms take three different classes of combinatorial structures and use them to generate .t; t/PH- planes and .t; t/AH-planes. Examples and properties of the combinatorial structures can be found in any standard work on combinatorial designs, e.g. [3, 4]. An Algorithm for Constructing Hjelmslev Planes 141

Definition 3.1 ([4, VII.2.2]). A projective plane is a set of points and lines such that • any two distinct points are incident with exactly one line. • any two distinct lines intersect at exactly one point. • there exist four points no three of which are on a common line. A projective plane of order m has m C 1 points on each line, m C 1 lines through each point, m2 C m C 1 points and m2 C m C 1 lines. A projective plane of order m may be represented as a 2 .m2 C m C 1; m C 1; 1/ block design [3, Defi 2.9]. Definition 3.2 ([4, VII.2.11]). A finite affine plane is a set of points and lines such that • any two distinct points are incident with exactly one line. • for any point P not on line l there is exactly one line through P that is parallel (has no points in common) with l. • there exist three points not on a common line. An affine plane of order m has m points on a line, m C1 lines through each point, m2 points and m2 C m lines. An affine plane can be obtained from a projective plane by removing one line and all the points on that line. An affine plane of order m may be represented as a 2.m2; m;1/block design [3, Defi 2.9]. An affine plane of order m may be partitioned into k-classes, with each k-class containing m lines. k-classes S and T are orthogonal if each line of S has exactly one element in common with each line of T. An affine plane of order m has m C 1 mutually orthogonal k-classes. Definition 3.3 ([4, III.3.5]). An orthogonal array OA.t; k;v/is a v2 k array with entries from a set of v symbols, such that in any t columns each ordered t-tuple occurs exactly once. Each symbol occurs in each column of the orthogonal array v times. An orthogonal array may be obtained from an affine plane by assigning each point to a row, and each k-class to a column of the array. The symbol in position i; j of the array indicates the line of k-class j that is incident with point i. Alternatively this structure may be considered as the dual of an affine plane. Thus a catalogue of affine planes also provides a catalogue of the appropriate orthogonal arrays. The three structures above may all be generated from a projective plane. However, it is not essential in the following algorithms that the objects have any relationship other than size. The construction of Drake and Shult [9]usesthe incidence matrix of a projective plane, for which each symbol is replaced by a sub-matrix which meets a set of conditions given by a semi-net with zings. The algorithm developed here is related to this method. 142 J.L. Hall and A. Rao

4 An Algorithm for Constructing 2-Uniform Projective Hjelmslev Planes

This algorithm is implicit in the proof of [10, Thm 1]. Algorithm 4.1. To create the structure H :

1. Let P be a projective plane of order m, let A0; A1;:::;Am2Cmbe a list of affine planes of order m and O0; O1;:::;Om2Cm be a list of orthogonal arrays OA.2; m C 1; m/. Note that some (or all) of the affine planes and some (or all) of orthogonal arrays may be the same. 2 2. For each point of P, replace point P with m points which are a copy of AP. This gives .m2 C m C 1/m2 points in H . Each affine plane will now be called a point-neighbourhood of H . 3. Choose a line g DfP0; P2;:::;Pmg; in P, and for each point of g choose a A ; A ;:::;A parallel class of each of P0 P1 Pm . Label each of the lines of the parallel class of each point-neighbourhood with the symbols from Og. Since each point- neighbourhood is in m lines of P, each time a particular point P of P is in a chosen line, a different parallel class of AP must be used. Label each column of Og with a point of g. 4. Each line of H is constructed by reading a row of Og and selecting the points which correspond to the lines of the point neighbourhoods. Repeat for each line of P. An example is given below. Step 1. A Projective plane of order 3, an Affine plane of order 3 and an orthogonal 2 array OA.2;4;3/. In this example Ai D A and Oi D O for all 0 Ä i Ä m C m. 8 9 ˆ 0; 1; 2; 9 ; > 8 9 ˆ f g > ˆ ; ; ; > ˆ 3; 4; 5; 9 ; > ˆ fR S Tg > ˆ f g > ˆ > ˆ > ˆ fU; V; Wg; > ˆ f6; 7; 8; 9g; > ˆ > LLLL ˆ > ˆ fX; Y; Zg; > ˆ f0; 3; 6; Ag; > ˆ > LMMM ˆ > ˆ fR; U; Xg; > ˆ f1; 4; 7; Ag; > ˆ > LNNN ˆ > ˆ fS; V; Yg; > <ˆ 2; 5; 8; ; => <ˆ => f Ag ; ; ; MLMN P 0; 4; 8; ; A fT W Zg O D ˆ f Bg > D ˆ ; ; ; > D MMN L ˆ 1; 5; 6; ; > ˆ fR V Zg > ˆ f Bg > ˆ > MN LM ˆ > ˆ fS; W; Xg; > ˆ f2;3;7;Bg; > ˆ > NLNM ˆ > ˆ fT; U; Yg; > ˆ f0; 7; 5; Cg; > ˆ > NMLN ˆ > ˆ fR; W; Yg; > ˆ f1; 3; 8; Cg; > ˆ > NNML ˆ > ˆ fS; U; Zg; > ˆ f2;4;6;Cg; > :ˆ ;> :ˆ ;> fT; V; Xg f9; A; B; Cg;

Step 2. The points of H can be written as a Cartesian product of the set of points in A and P. An Algorithm for Constructing Hjelmslev Planes 143

f.0; R/; .0; S/; .0; T/; .0; U/; .0; V/; .0; W/; .0; X/;:::;.C; X/; .C; Y/; .C; Z/g:

Step 3. Choosing line g Df3; 4; 5; 9g, the chosen k-classes of each point neighbourhood of g, and the labels for the columns of O. In neighbourhood 3; L WD fR; S; Tg, M WD fU; V; Wg, N WD fX; Y; Zg. In neighbourhood 4; L WD fR; S; Tg, M WD fU; V; Wg, N WD fX; Y; Zg. In neighbourhood 5; L WD fR; S; Tg, M WD fU; V; Wg, N WD fX; Y; Zg. In neighbourhood 9; L WD fR; U; Xg, M WD fS; V; Yg, N WD fT; W; Zg.

3459 LLLL L MMM LNNN MLMN MMN L MN LM NLNM NMLN NNML

Step 4. The lines of H in the line-neighbourhoods corresponding to the line f3; 4; 5; 9g of P are constructed according to O. Note that every pair of lines from within a line neighbourhood shares exactly 3 points.

f.3; R/; .3; S/; .3; T/; .4; R/; .4; S/; .4; T/; .5; R/; .5; S/; .5; T/; .9; R/; .9; U/; .9; X/g f.3; R/; .3; S/; .3; T/; .4; U/; .4; V/; .4; W/; .5; U/; .5; V/; .5; W/; .9; S/; .9; V/; .9; Y/g f.3; R/; .3; S/; .3; T/; .4; X/; .4; Y/; .4; Z/; .5; X/; .5; Y/; .5; Z/; .9; T/; .9; W/; .9; Z/g f.3; U/; .3; V/; .3; W/; .4; R/; .4; S/; .4; T/; .5; U/; .5; V/; .5; W/; .9; T/; .9; W/; .9; Z/g f.3; U/; .3; V/; .3; W/; .4; U/; .4; V/; .4; X/; .5; X/; .5; Y/; .5; Z/; .9; R/; .9; U/; .9; X/g f.3; U/; .3; V/; .3; W/; .4; X/; .4; Y/; .4; Z/; .5; R/; .5; S/; .5; T/; .9; S/; .9; V/; .9; Y/g f.3; X/; .3; Y/; .3; Z/; .4; R/; .4; S/; .4; T/; .5; X/; .5; Y/; .5; Z/; .9; S/; .9; V/; .9; Y/g f.3; X/; .3; Y/; .3; Z/; .4; U/; .4; V/; .4; X/; .5; R/; .5; S/; .5; T/; .9; T/; .9; W/; .9; Z/g f.3; X/; .3; Y/; .3; Z/; .4; X/; .4; Y/; .4; Z/; .5; U/; .5; V/; .5; W/; .9; R/; .9; U/; .9; X/g

Theorem 4.2. The structure generated by Algorithm 4.1 is a 2-uniform .m; m/PH- plane. Proof. This algorithm generates an incidence structure H with .m2 C m C 1/m2 points, .m2 C m C 1/m2 lines, each line containing .m2 C m/ points, and each point incident with .m2 C m/ lines. Axioms 1 and 4: Any pair of points P and Q which are in the same point- neighbourhood are incident with exactly one line of the point neighbourhood restriction, which is an afﬁne plane. Each line of the point neighbourhood restriction is used in m lines of H , as each symbol appears m times in each column of O.For points P and R which are in different point-neighbourhoods, there is exactly one line 144 J.L. Hall and A. Rao

of P which is incident with any pair of point-neighbourhoods. Given k-classes PN X and RN Y of each point-neighbourhood, O ensures that each line of PN X is in a line of H with each line of RN Y exactly once. Axioms 2 and 3: O ensures that lines in the same line neighbourhood intersect in exactly one line of a k-class of a point-neighbourhood, which is m points. For lines g and h which are in different line-neighbourhoods, their line-neighbourhoods may be labeled with lines from P. Any pair of lines in P intersect in exactly one point, thus any line-neighbourhoods of H intersect in exactly one point neighbourhood QN . Each line-neighbourhood is allocated a different k-class QN X, QN Y . Thus the line g in H must contain a line of QN X and h a line of QN Y .Asthek-classes are orthogonal, g and h intersect in exactly one point. Let collapse point-neighbourhoods and line neighbourhoods. It is trivial to check that this is incidence preserving and surjective, and thus an epimorphism. All axioms of Definition 2.1 are satisfied. Thus H is a projective Hjelmslev plane. The cardinalities of H show that H is an .m; m/PH-plane and is therefore 2- uniform. In the example the affine plane used is a sub-geometry of the given projective plane. However this is not required and any affine planes of the appropriate size may be used. Any projective plane, any affine planes and any orthogonal arrays of the appropriate sizes may be used. Theorem 4.3. All 2-uniform projective Hjelmslev planes can be generated using Algorithm 4.1. Proof. From Theorem 4.2, H is a 2-uniform projective Hjelmslev plane. Axiom 1 of Definition 2.4 requires that point-neighbourhood restrictions are affine planes. This is guaranteed by step 2. Requiring that every line of every PN is the restriction of the same number of lines is equivalent to ensuring that each line of each parallel class of the point-neighbourhood is included in the same number of lines at step 4. This is ensured as each symbol occurs in each column of an orthogonal array the same number of times.

5 An Algorithm for Constructing 2-Uniform Afﬁne Hjelmslev Planes

Algorithm 4.1 may be amended to construct 2-uniform afﬁne Hjelmslev planes Algorithm 5.1. To create the structure H :

1. Let A be an affine plane of order m, let A0; A1;:::;Am2 be a list of affine planes of order m and O0; O1;:::;Om2Cm be a list of orthogonal arrays OA.2; m; m/. Note that some (or all) of the affine planes and orthogonal arrays may be the same. An Algorithm for Constructing Hjelmslev Planes 145

2. For each point of A , replace point P with m2 points which are a copy of 4 AP. This gives m points in H . Each affine plane will now be called a point- neighbourhood of H . 3. Choose a line g DfP0; P2;:::;Pm1g; in H , and for each point of g choose A ; A ;:::;A a parallel class of each of P0 P1 Pm1 . Label each of the lines of the parallel class of each point-neighbourhood with the symbols from Og. Since each point-neighbourhood is in m lines of P, each time a particular point P of P is in a chosen line, a different parallel class of AP must be used. Label each column of Og with a point of g. 4. Each line of A is constructed by reading a row of Og and selecting the points which correspond to the lines of the point neighbourhoods. Repeat for each line of H . Theorem 5.2. All 2-uniform affine Hjelmslev planes can be generated using Algo- rithm 5.1. The proof is similar to that of Theorem 4.2 and is omitted. Affine planes are sub-geometries of projective planes, the same is sometimes true for affine Hjelmslev planes. Lemma 5.1 ([8]). A .t; r/PH-plane can be truncated to a .t; r/AH-plane. Proof. Take a .t; r/PH-plane and remove all the lines of one line-neighbourhood together with all incident points. However not all affine Hjelmslev planes may be extended to a projective Hjelmslev plane [1]. As a corollary of Theorems 4.2 and 5.2 we have a new proof that .t; t/AH-planes are well behaved in this respect. Corollary 5.1 ([1]). All .t; t/AH-planes are a sub-geometry of a .t; t/PH-plane. Proof. All orthogonal arrays OA.2; m; m/ are extendible to OA.2; m C 1; m/ [4, §III Thm 3.95], and hence all .t; t/AH-planes are extendible to .t; t/PH-planes.

6 Conclusions

For orders where there are several possible projective planes, afﬁne planes and orthogonal arrays, these algorithms generate many different Hjelmslev planes of the same size. The problem of determining if two planes are isomorphic is also computational intensive. Even with the same seeds, there are an enormous number of choices to be made in step 3, resulting in an explosion of the number of planes generated. With no cut downs, .m2 C m C 1/Š.m C 1/ŠmŠmŠ projective Hjelmslev planes can be generated using Algorithm 4.1. When there is an algebraic structure on the plane, the automorphism group can be very large [15]. Thus a large number of the planes generated by this algorithm are isomorphic. 146 J.L. Hall and A. Rao

Further analysis of the choices made in step 3 may reduce the number of isomorphic planes generated. Some ground breaking work is needed on automorphisms of Hjelmslev planes to reduce this to a sub-factorial number. As mentioned before there has been little research on isomorphisms of Hjelmslev planes. There are some results concerning isomorphism classes of Hjelmslev planes of uniformity 3 or greater [10], however isomorphisms of 2-uniform Hjelmslev planes appear to be a more difficult problem. This algorithm has been implemented in Python by Jesse Waechter-Cornwill with a visual interpretation. See http://joannelhall.com/gallery/hjelmslev/ There is also scope for extending this algorithm to affine Hjelmslev planes of higher uniformity by using the 2-uniform affine Hjelmslev plane generated in Algorithm 5.1, and the corresponding array structure as inputs in Algorithm 4.1.

Acknowledgements Thanks to Jesse Waechter-Cornwill for the coding and visualisation of Algorithm 4.1. Thanks are due to Cathy Baker for highlighting reference [10].

References

1. Bacon, P.Y.: On the extension of projectively uniform affine Hjelmslev planes. Abh. Math. Sem. Hamburg 41(1), 185–189 (1974) 2. Bailey, R., Cameron, P.J., Dobcsányi, P., Morgan, J.P., Soicher, L.H.: DesignTheory.org. U.K. Engineering and Physical Sciences Research Council (Updated 2012). designtheory.org 3. Beth, T., Jungnickel, D., Lenz, H.: Design Theory, vol. 69. Cambridge University Press, Cambridge (1999) 4. Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs. CRC Press, Boca Raton (2010) 5. Craig, R.T.: Extensions of finite projective planes. I. Uniform Hjelmslev planes. Can. J. Math 16, 261–266 (1964) 6. Dembowski, P.: Finite Geometries. Classics in Mathematics, vol. 44. Springer, Berlin (1996) 7. Drake, D.A.: On n- uniform Hjelmslev planes. J. Comb. Theory 9(3), 267–288 (1970) 8. Drake, D.A.: Nonexistence results for finite Hjelmslev planes. Abh. Math. Sem. Hamburg, 40(1), 100–110 (1974) 9. Drake, D.A., Shult, E.E.: Construction of Hjelmslev planes from (t, k)-nets. Geom. Dedicata 5(3), 377–392 (1976) 10. Hanssens, G., Van Maldeghem, H.: A universal construction for projective Hjelmslev planes of level n. Compos. Math. 71(3), 285–294 (1989) 11. Hjelmslev, J.: Die Geometrie der Wirklichkeit. Acta Math. 40(1), 35–66 (1916) 12. Honold, T., Kiermaier, M.: The existence of maximal .q2;2/-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic. Design Code Cryptogr. 68(1- 3), 105–126 (2013) 13. Honold, T., Landjev, I.: On arcs in projective Hjelmslev planes. Discret. Math. 231(1), 265–278 (2001) 14. Honold, T., Landjev, I.: Non-free extensions of the simplex codes over a chain ring with four elements. Design Code Crypt. 66(1–3), 27–38 (2013) 15. Kiermaier, M., Koch, M., Kurz, S.: 2-arcs of maximal size in the affine and the projective Hjelmslev plane over Z25. Adv. Math. Commun. 5(2), 287–301 (2011) An Algorithm for Constructing Hjelmslev Planes 147

16. Kiermaier, M., Zwanzger, J.: New ring-linear codes from dualization in projective Hjelmslev geometries. Design. Code. Crypt. 66(1–3), 39–55 (2013) 17. Kleinfeld, E. : Finite Hjelmslev planes. Illinois J. Math. 3(3), 403–407 (1959) 18. Klingenberg, W.: Projektive und afﬁne Ebenen mit Nachbarelementen. Math. Z. 60(1), 384– 406 (1954) 19. Saniga, M., Planat, M.: Hjelmslev geometry of mutually unbiased bases. J. Phys. A Math. Gen. 39(2), 435 (2006) 20. Sloane., N.J.A.: A library of orthogonal arrays. http://neilsloane.com/oadir/ 21. Veldkamp, F.D.: Geometry over rings. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, pp. 1033–1084. Elsevier, Amsterdam (1995) Mutually Unbiased Biangular Vectors and Association Schemes

W.H. Holzmann, H. Kharaghani, and S. Suda

Abstract A class of unbiased .1; 1/-matrices extracted from a single Hadamard matrix is shown to provide uniform imprimitive association schemes of four class and six class.

Keywords Biangular vectors • Unbiased Hadamard matrices • Association schemes

1 Introduction

Let S be a set of unit vectors in Rn and 0 Ä ˛<1, 0 Ä ˇ<1be two distinct real numbers. S is called a .˛; ˇ/-biangular set if jhu;vij belongs to f˛; ˇg for every distinct pair u;v 2 S. Two matrices of order n are called .˛; ˇ/-unbiased if the collection of their normalized row vectors forms an .˛; ˇ/-biangular set in Rn. A set of matrices is called mutually .˛; ˇ/-unbiased if every pair of matrices in the set is .˛; ˇ/- unbiased. A Hadamard matrix is a square .1; 1/-matrix with mutually orthogonal rows. If it also has constant row (or column) sum, then it is said to be regular. Mutually unbiased Hadamard (MUH) matrices form a special subset of unbiased biangular matrices. They are Hadamard matrices of order n such that the absolute value of the inner product of normalized rows of distinct matrices is all equal to

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. W.H. Holzmann • H. Kharaghani () Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, Canada T1K 3M4 e-mail: [email protected]; [email protected] S. Suda Department of Mathematics Education, Aichi University of Education, Kariya, Aichi 448-8542, Japan e-mail: [email protected]

© Springer International Publishing Switzerland 2015 149 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_12 150 W.H. Holzmann et al.

p 1= n. MUH matrices, leading to mutually unbiased bases (MUB) are of much interest and are being studied extensively, see [2–4, 7, 9, 13, 16]. A .1; 1/-matrix of order n is called .˛; ˇ/-biangular if its normalized rows form an .˛; ˇ/-biangular set in Rn.An.˛; ˇ/-biangular matrix H of order nm is called regular if the rows of H can be partitioned into m-classes of size n each in such a way that: 1. jhu;vij D ˛ for each distinct pair u;vin the same class, 2. jhu;vij D ˇ for each pair u;vbelonging to different classes. A symmetric d-class association scheme,see[1, 10], with vertex set X of size n and d classes is a set of symmetric .0; 1/-matrices A0;:::;Ad, which are not equal to zero matrix, with rows and columns indexed by X, such that:

1. PA0 D In, the identity matrix of order n. d 2. iD0 Ai D Jn, the allP ones matrix of order n. d k k 3. For all i, j, AiAj D kD0 pijAk for some pij’s.

It follows from property (3) that the Ai’s necessarily commute. The vector space spanned by Ai’s forms a commutative algebra, denoted by A and called the Bose– Mesner algebra or adjacency algebra. There exists a basis of A consisting of primitive idempotents, say E0 D .1=n/Jn; E1;:::;Ed.SincefA0; A1;:::;Adg and fE0; E1;:::;Edg are two bases of A , there exist the change-of-bases matrices . /d . /d P D Pij i;jD0, Q D Qij i;jD0 so that

Xd 1 Xd A D P E ; E D Q A : j ij j j n ij j iD0 iD0

Since disjoint .0; 1/-matrices Ai’s form a basis of A , the algebra A is closed under the entrywise multiplication denoted by ı.TheKrein parameters qk ’s are defined P ij 1 d k . k /d by Ei ı Ej D n kD0 qijEk.TheKrein matrix Bi is defined as Bi D qij j;kD0. Each of the matrices Ai’s can be considered as the adjacency matrix of some graph without multiedges. The scheme is imprimitive if, on viewing the Ai’s as adjacency matrices of graphs Gi on vertex set X, at least one of the Gi’s, i ¤ 0, is disconnected. In thisP case, there exists a set I of indices such that 0 and such i I ; < are elements of and j2I Aj D Ip ˝ Jq for some p q with p n. Thus the set of n vertices X are partitioned into p subsets called fibers, each of which has size q.The I 0;1;:::; k 0 set defines an equivalence relation on f dg by j k if and only if pij ¤ for some i 2 I .LetI0 D I ; I1;:::;It be the equivalent classes on f0;1;:::;dg by .Thenby[1, Theorem 9.4] there exist .0; 1/-matrices Aj (0 Ä j Ä t) such that X Ai D Aj ˝ Jq ;

i2Ij and the matrices Aj (0 Ä j Ä t) deﬁne an association scheme on the set of ﬁbers. This is called the quotient association scheme with respect to I . Schemes from Hadamard Matrices 151

For ﬁbers U and V,letI .U; V/ denote the set of indices of adjacency matrices .0; 1/ UV that has an edge between U and V.Wedeﬁnea -matrix Ai by ( 1 . / 1; ; ; . UV / if Ai xy D x 2 U y 2 V Ai xy D 0 otherwise:

Deﬁnition 1.1. An imprimitive association scheme is called uniform if its quotient k association scheme is class 1 and there exist integers aij such that for all ﬁbers U; V; W and i 2 I .U; V/; j 2 I .V; W/,wehave X UV VW k UW : Ai Ai D aijAk k

It is easy to see that only Hadamard matrices of square order n can form a class of MUH matrices. Any class of mutually unbiased regular Hadamard matrices forms a system of linked designs deﬁned by Cameron in [5]. Mathon [14]showed that a system of linked designs is equivalent to a uniform imprimitive 3-class association scheme. There were only two known classes of systems of linked designs, see [15], until recently when a new method of construction for MU regular Hadamard matrices was demonstrated in [11] leading to more systems of linked designs. In this paper we remove the assumption of the existence of a system of linked designs and replace it with a weaker condition. We present a large class of mutually unbiased .0; ˇ/-biangular matrices which lead to new classes of uniform imprimitive 4-class and 6-class association schemes. We will observe that if the matrices in the class are all Hadamard, then we have the same class of uniform imprimitive 3-class association schemes introduced by Mathon in [14].

2 A Class of Regularly Biangular Matrices

We begin with a lemma from [12]. Lemma 2.1. Let H be a normalized Hadamard matrix of order n. Then there are symmetric .1; 1/-matrices C1; C2;:::;Cn such that:

1. C1 D Jn. 2. CiCj D 0, 1 Ä i ¤ j Ä n. 2 1 3.P Ci D nCi, Ä i Ä n. n 4. iD1 Ci D nIn.

It follows from these conditions that the row sums and column sums are 0 for Ci, i ¤ 1, and that 152 W.H. Holzmann et al.

Xn 2 2 : Ci D n In nJn iD2

t 1;2;:::; Proof. Let ri be the i-th row of H. Consider Ci D ri ri, i D n. Lemma 2.2. Let n be the order of a Hadamard matrix. Then there is a regular .0; 1=.n 1//-biangular matrix of order n.n 1/.

Proof. Starting with a Latin square on the set f2;:::;ng and substituting i with Ci from Lemma 2.1,fori D 2;:::;n, we obtain a matrix which we will denote by L. Clearly L is a .1; 1/-matrix of order n.n 1/. It follows from Lemma 2.1 that LLt is the diagonal matrix with all diagonal blocks equal to:

2 2 2 2 : C2 C C3 CCCn D n In nJn

Let Ri be the i-th block row of L of size n.Thenjhu;vij D n for each pair of row vectors u, v in the same Ri, while jhu;vij D 0 for each u and v from different Ri’s. This completes the proof.

Remark 2.1. The row and column sum of each of the Ci’s, for i D 2;:::;n in the preceding Lemma’s proof is zero, so the same is true for L. The following lemma is immediate from Lemma 2.1. . 2 2 2/ 2 2;3;:::; Lemma 2.3. C2 C C3 CCCn Ci D n Ci,foreachiD n.

Two Latin squares L1 and L2 of size n on symbol set f0;1;2;:::;n1g are called suitable if every superimposition of each row of L1 on each row of L2 results in only one element of the form .a; a/. A set of Latin squares in which every distinct pair of Latin squares is mutually suitable is called mutually suitable latin squares, denoted MSLS. Lemma 2.4 ([11, Lemma 9]). There exist m MOLS (Mutually Orthogonal Latin Squares) of size n if and only if there exist m MSLS of size n. Theorem 2.1. Let n be the order of a normalized Hadamard matrix H. Let ` be the 1 ` 1 .0; 1 / number of MSLS of order n . Then there are C mutually n1 -unbiased regularly biangular matrices of order n.n 1/. Proof. Starting with ` Latin squares of size n 1 on the set f2;:::;ng and substituting i with Ci from Lemma 2.1,fori D 2;:::;n,letfL1;:::;L`g be the 1 1;:::; ` ` .0; / resulting matrices. It follows that fL L g forms a set of mutually n1 - unbiased regularly biangular matrices of order n.n 1/. These are ` mutually unbiased regularly biangular matrices of order n.n 1/.To obtain an additional one, let ri be the i-th row of H, i D 2;3;:::;n.LetK D Œkij D t . 1/ .0; 1 / rjri ,thenK is a regularly biangular matrix of order n n which is n1 - unbiased with Li, i D 1;2;:::;`. Schemes from Hadamard Matrices 153

Remark 2.2. Let n be the order of a Hadamard matrix. According to Lemma 2.1, there are n auxiliary matrices which can be used in Latin squares of size n to construct mutually unbiased regular Hadamard matrices of order n2,see[11]. We have removed the matrix C1, the all ones matrix, and used Latin squares of size n1 here. As a result, the matrices found are a little short of being Hadamard, but they .0; 1 / are mutually n1 -unbiased. Note that there is a complete set of MSLS whenever n 1 is a prime power. This is a great gain in the number of biangular vectors. We will show in the next section that these matrices may be used to construct 4-class and 6-class association schemes. Remark 2.3. If we replace 1 with 0 in the matrix constructed in Theorem 2.1,then the row vectors form a constant weight code of length n.n 1/, weight n.n 1/=2 and minimum distance n.n 2/=2. By adding the all-ones vectors to each codeword n.n1/ in Z2 , the number of codewords is 2.` C 1/n.n 1/.

3 A Class of Mutually Unbiased Regularly Biangular Matrices and Association Schemes

n Let S Dfv1;v2;:::;vsg be an .˛; ˇ/-biangular set of s vectors in R .LetG be the Gram matrix of S, where the rows and columns are indexed with the given ordering on S. Then the diagonal entries of G are all one and off-diagonal entries belong to the set f˛; ˛; ˇ; ˇg. There are a number of ways to break G into disjoint .0; 1/- matrices and search for the possibility of different classes of association schemes. We will show in this section some examples where such an attempt is fruitful. Theorem 3.1. Let n be the order of a Hadamard matrix. Let ` be the number of MSLS of order n 1. Then the Gram of any subset consisting of m of the ` 1 .0; 1 / . 1/ C mutually n1 -unbiased regularly biangular matrices of order n n constructed in Theorem 2.1 form an imprimitive 4-class association scheme.

Proof. Let H1; H2;:::;Hm be any subset of ` C 1 matrices constructed in Theo- rem 2.1. Let 2 3 H1 6 7 6 H2 7 G D 6 : 7 Ht Ht ::: Ht : 4 : 5 1 2 m

Using the properties of the given matrices, we can write

2 G D Im ˝ In1 ˝ .n In nJn/ C nN ; 154 W.H. Holzmann et al. where N is a .1; 1/-matrix. Using Lemmas 2.2 and 2.3, we see that G2 D mn2G. Write N D NC N,whereNC and N are disjoint .0; 1/-matrices. Let

A0 D Imn.n1/ ;

A1 D Im.n1/ ˝ .Jn In/; C A2 D N ; A3 D N ;

A4 D Im ˝ Jn.n1/ Im.n1/ ˝ Jn :

Then A0 C A1 C A2 C A3 C A4 D Jmn.n1/.Furthermore,

A1A1 D .n 1/A0 C .n 2/A1 ; n 2 n A1A2 A2 A3 ; D 2 C 2 n n 2 A1A3 A2 A3 ; D 2 C 2

A1A4 D .n 1/A4 ; n.n 1/.m 1/ n.n 2/.m 1/ n2.m 2/ A2A2 A0 A1 A2 D 2 C 4 C 4 n.n 2/.m 2/ n.n 1/.m 1/ A3 A4 ; C 4 C 4 n2.m 1/ n.n 2/.m 2/ n2.m 2/ n.n 1/.m 1/ A2A3 A1 A2 A3 A4 ; D 4 C 4 C 4 C 4 n.n 2/ n.n 2/ A2A4 A2 A3 ; D 2 C 2 n.n 1/.m 1/ n.n 2/.m 1/ n2.m 2/ A3A3 A0 A1 A2 D 2 C 4 C 4 n.n 2/.m 2/ n.n 1/.m 1/ A3 A4 ; C 4 C 4 n.n 2/ n.n 2/ A3A4 A2 A3 ; D 2 C 2 2 2 2 A4A4 D .n 2n/A0 C .n 2n/A1 C .n 3n/A4 :

Since the graph associated with the matrix A4 is a disjoint union of cliques of size n, it follows that A1, A2, A3, A4 form an imprimitive 4-class association scheme. Schemes from Hadamard Matrices 155

Remark 3.1. This imprimitive association scheme is uniform in the sense of van Dam et al. [6]. The ﬁrst and the second eigenmatrices [1] are as follows. 0 1 . 1/. 1/ . 1/. 1/ 1 n 1 n n m n n m n.n 2/ B 2 2 C B1 1 n.m1/ n.m1/ 0 C B 2 2 C B1 10 0 C ; P D B n n C @ n n A 1 1 2 2 0 n.n1/ n.n1/ 1 n 1 2 2 n.n 2/ 0 1 1.n 1/2 .n 2/m .n 1/2.m 1/ m 1 B C B1 n 1.n 2/m .n 1/.m 1/ m 1C B C C B1 10 1 1 C : Q D B n n C C @1 n C 10 n 1 1 A 10 m 0 m 1

. k /4 With the above ordering of primitive idempotents, the Krein matrix B4 D q4;j j;kD0 is the following form: 0 1 00001 B C B 00010C B C B 00 10 0C : B4 D B m C @ 0 m 10m 20A m 10 0 0m 2

Thus this association scheme certainly satisﬁes [6, Proposition 4.7]. Furthermore we obtain the association scheme of class 6 from the association scheme in Theorem 3.1.

Theorem 3.2. Let A0,A1, :::,A4 be the adjacency matrices of the association scheme in Theorem 3.1. Deﬁne Â Ã Â Ã Â Ã Â Ã A0 0 A1 0 A2 A3 A3 A2 AQ 0 D ; AQ 1 D ; AQ 2 D ; AQ 3 D ; 0 A0 0 A1 A3 A2 A2 A3 Â Ã Â Ã Â Ã A4 A4 0 A0 0 A1 AQ 4 D ; AQ 5 D ; AQ 6 D : A4 A4 A0 0 A1 0

Then AQ 0, :::, AQ 6 form an association scheme. Proof. Follows from the calculation in Theorem 3.1. Remark 3.2. The association scheme in Theorem 3.2 is also uniform. The ﬁrst and second eigenmatrices and B6 are as follows: 156 W.H. Holzmann et al. 0 1 1 n 1 n.n 1/.m 1/ n.n 1/.m 1/ 2n.n 2/ 1 n 1 B C B1 1 n.m 1/ n.m 1/ 0 11C B C B1 n 10 0 2n 1 n 1 C B C B1 10 0 011 C ; P D B C B1 10 0 01 1C B n n C C @1 1 nn0 11A 1 n 1 n.n 1/ n.n 1/ 2n.n 2/ 1 n 1 0 1 1.n 1/2 .n 2/m .n 1/2m .n 1/m .n 1/2.m 1/ m 1 B C B1 n C 1.n 2/m .n 1/m .n 1/m .n 1/.m 1/ m 1C B C B1 n 10 0 0 n 1 1 C B C C B1 10 0 0 1 1 C ; Q D B n C n C B10 00 0 1C B m m C @1 .n 1/2 .n 2/m .n 1/2m .n 1/m .n 1/2.m 1/ m 1A 1 n 1.n 2/m .n 1/m .n 1/m .n 1/.m 1/ m 1 0 1 0000001 B C B 0000010C B C B 00m 10000C B C B 000 1000C : B6 D B m C B 0000 10 0C B m C @ 0 m 10 0 0m 20A m 100000m 2

Thus the association scheme also certainly satisﬁes [6, Proposition 4.7]. Remark 3.3. Referring to Theorem 2.1, if the Latin squares of size n on the set f1;2;:::, ng and the matrices C1; C2;:::;Cn of Lemma 2.1 are used, then the ` C 1 matrices of Theorem 2.1 are mutually unbiased regular Hadamard matrices. Furthermore, the classes A1 and A4 in Theorem 3.1 reduce to the identity matrix and Jn2 In2 , respectively. Thus we have an imprimitive 3-association scheme. This fact was shown quite elegantly in [14] to be equivalent to a system of linked designs. This seems to be quite interesting, as each of the matrices in Theorem 2.1 can be made (by replacing all negative entries by zero) to be a group divisible design,see [8] for the deﬁnition. This suggests that the .0; 1/-unbiased matrices of Theorem 3.1 form what we wish to call a system of linked group divisible designs. We hope to investigate this concept further in a future work.

Acknowledgements Kharaghani thanks NSERC for the continuing support of his research. Schemes from Hadamard Matrices 157

References

1. Bannai, E., Ito, T.: Algebraic Combinatorics. I. Association Schemes. The Ben- jamin/Cummings Publishing Co., Inc., Menlo Park (1984) 2. Best, D., Kharaghani, H.: Unbiased complex Hadamard matrices and bases. Cryptogr. Com- mun. 2(2), 199–209 (2010) 3. Boykin, P.O., Sitharam, M., Tariﬁ, M., Wocjan, P.: Real mutually unbiased bases. Preprint. arXiv:quant ph/0502024v2 4. Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. Lond. Math. Soc. (3) 75(2), 436–480 (1997) 5. Cameron, P.J.: On groups with several doubly-transitive permutation representations. Math. Z. 128, 1–14 (1972) 6. van Dam, E.R., Martin, W.J., Muzychuk, M.: Uniformity in association schemes and coherent conﬁgurations: cometric Q-antipodal schemes and linked systems. J. Comb. Theory Ser. A 120(7), 1401–1439 (2013) 7. Durt, T., Englert, B.H., Bengtsson, I., Zyczkowski,˙ K.: On mutually unbiased bases. Int. J. Quantum Inf. 8, 535–640 (2010) 8. Ge, G.: Group divisible designs. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combina- torial Designs. Discrete Mathematics and Its Applications (Boca Raton), 2nd edn, pp. xxii+984. Chapman & Hall/CRC, Boca Raton (2007) 9. Godsil, C., Roy, A.: Equiangular lines, mutually unbiased bases, and spin models. Eur. J. Comb. 30(1), 246–262 (2009) 10. Godsil, C.D., Song, S.Y.: Association schemes. In: Colbourn, C.J., Dinitz, J.H. (eds.) Hand- book of Combinatorial Designs, 2nd edn, pp. 325–330. Chapman & Hall/CRC, Boca Raton (2007) 11. Holzmann, W.H., Kharaghani, H., Orrick, W.: On the real unbiased Hadamard matrices. In: Combinatorics and Graphs. Contemporary Mathematics, vol. 531, pp. 243–250. American Mathematical Society, Providence, RI (2010) 12. Kharaghani, H.: New class of weighing matrices. Ars Comb. 19, 69–72 (1985) 13. LeCompte, N., Martin, W.J., Owens, W.: On the equivalence between real mutually unbiased bases and a certain class of association schemes. Eur. J. Comb. 31(6), 1499–1512 (2010) 14. Mathon, R.: 3-class association schemes. In: Proceedings of the Conference on Algebraic Aspects of Combinatorics (University Toronto, Toronto, Ontario, 1975), pp. 123–155. Con- gressus Numerantium, No. XIII. Utilitas Math., Winnipeg, Man. (1975) 15. Mathon, R.: The systems of linked 2 .16; 6; 2/ designs. Ars Comb. 11, 131–148 (1981) 16. Wocjan, P., Beth, T.: New construction of mutually unbiased bases in square dimensions. Quantum Inf. Comput. 5(2), 93–101 (2005) A Simple Construction of Complex Equiangular Lines

Jonathan Jedwab and Amy Wiebe

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract A set of vectors of equal norm in Cd represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is d2, and it is conjectured that sets of this maximum size exist in Cd for every d 2. We describe a new construction for maximum- sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.

Keywords Combinatorial design theory • Complex equiangular lines • Hadamard matrix • Innner product

1 Introduction

Equiangular lines have been studied for over 65 years [13], and their construction remains “[o]ne of the most challenging problems in algebraic combinatorics” [16]. In particular, the study of equiangular lines in complex space has intensiﬁed recently, as its importance in quantum information theory has become apparent [1, 9, 17, 18]. The main question regarding complex equiangular lines is whether the well-known upper bound (see [6], for example) on the number of such lines is attainable in all dimensions: that is, whether there exist d2 equiangular lines in Cd for all integers d 2. Zauner [19] conjectured 15 years ago that the answer is yes. This conjecture is supported by exact examples in dimensions 2, 3 [5, 17], 4, 5 [19],

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. J. Jedwab ()•A.Wiebe Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6 e-mail: [email protected]; [email protected]

© Springer International Publishing Switzerland 2015 159 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_13 160 J. Jedwab and A. Wiebe

6[8], 7 [1, 16], 8 [4, 9, 14, 18], 9–15 [9–11], 16 [3], 19 [1, 16], 24 [18], 28 [4], 35and48[18], and by examples with high numerical precision in all dimensions d Ä 67 [17, 18]. Hoggar [14] gave a construction for d D 8 in 1981. Although other examples in C8 have since been found [4, 9, 18], Hoggar’s 64 lines are simplest to construct and can be interpreted geometrically as the diameters of a polytope in quaternionic space. Appleby [2] observed in 2011: “In spite of strenuous attempts by numerous investigators over a period of more than 10 years we still have essentially zero insight into the structural features of the equations [governing the existence of a set of d2 equiangular lines in Cd] which causes them to be soluble. Yet one feels thattheremustsurelybesuchastructuralfeature...(oneofthefrustratingfeatures of the problem as it is currently formulated is that the properties of an individual [set of d2 equiangular lines in Cd] seem to be highly sensitive to the dimension).” In view of this, finding a structure for maximum-sized sets of equiangular lines that is common across multiple dimensions is highly desirable. One such example, due to Khatirinejad [16], links dimensions 3, 7 and 19 by constraining the “fiducial vector,” from which the lines are constructed, to have all entries real. In this paper we construct a maximum-sized set of equiangular lines in Cd from an order d Hadamard matrix, for d 2f2;3;8g. This gives a new connection between the study of complex equiangular lines and combinatorial design theory. It also links three different dimensions d. Although we show that the construction in its current form cannot be extended to other values of d, we speculate that the construction could be modified to deal with other dimensions by using more than one complex Hadamard matrix. The constructed set of 64 equiangular lines in C8 is of particular interest. It is almost flat: all but one of the components of each of its 64 lines have equal magnitude. It turns out (Zhu, 2014, personal communication) that this constructed set is equivalent to Hoggar’s 64 lines under a transformation involving the unitary matrix 0 1 00 i 10 0 1 i B C B 001 i 00 i 1 C B C B 1 i 00i 100C B C 1 B i 1001 i 00C U B C : D 2 B 00 10 0 1 C (1) B i i C B C B 001i 00 i 1 C @ 1 i 00i 100A i 1001 i 00

x C8 (Speciﬁcally, the Hoggar lines are equivalent to a set f pjg of 64 lines in given by Godsil and Roy [7, p. 6], and our constructed set is f 2Uxjg.) We believe that the description of the Hoggar lines presented here, using the basis obtained from a Hadamard matrix of order 8, is simpler than Hoggar’s construction. A Simple Construction of Complex Equiangular Lines 161

2 Complex Equiangular Lines from Hadamard Matrices

We now introduce the main objects of study. A line through the origin in Cd can be represented by a nonzero vector x 2 Cd which spans it. The angle between two lines in Cd represented by vectors x; y is given by Â Ã jhx; yij arccos ; jjxjj jjyjj p where hx; yi is the standard Hermitian inner product in Cd and jjxjj D jhx; xij is d the norm of x.Asetofm 2 distinct lines in C , represented by vectors x1;:::;xm, is equiangular if there is some real constant c such that Â Ã jhx ; x ij arccos j k D c for all j ¤ k: jjxjjj jjxkjj

To simplify notation, we can always take x1;:::;xm to have equal norm, and then it sufﬁces that there is a constant a such that

jhxj; xkij D a for all j ¤ k:

An order d complex Hadamard matrix is a d d matrix, all of whose entries are in C and are of magnitude 1, for which

HH D dId; where H is the conjugate transpose of H (so that the rows of H are pairwise orthogonal). If, additionally, the entries of H are all in f1; 1g,thenH is called a real Hadamard matrix or just a Hadamard matrix. A simple necessary condition for the existence of a Hadamard matrix of order d >2is that 4 divides d; it has long been conjectured that this condition is also sufﬁcient (see [15], for example). We call two complex Hadamard matrices H; H0 equivalent if there exist diagonal unitary matrices D; D0 and permutation matrices P; P0 such that

H0 D DPHP0D0:

Example 2.1. Let ! D e2i=3 and let H; H0 be the order 3 complex Hadamard matrices: 0 1 0 1 11 1 !11 H D @ 1!!2 A H0 D @ 1!1A : 1!2 ! !2 !2 1 162 J. Jedwab and A. Wiebe

Then H and H0 are equivalent, since we can obtain H0 from H by interchanging columns 2 and 3 and then multiplying the resulting ﬁrst column and third row by ! and the resulting second row by !2.Thatis, 0 1 0 1 0 1 0 1 100 100 100 !00 H0 D @ 0!2 0 A @ 010A H @ 001A @ 010A 00! 001 010 001

D DPHP0D0:

We now describe the main construction of this paper. Let H be an order d complex Hadamard matrix. Consider H to represent d vectors given by the rows of the matrix. Form d sets of d vectors H1.v/; : : : ; Hd.v/ from H,whereHj.v/ is the set of vectors formed by multiplying coordinate j of each vector of H by the v C .v/ d .v/ constant 2 ,andletH D[jD1Hj . The main result of this paper, given in Theorem 3.1, is that in dimensions d D 2;3;8 we can construct a set of d2 equiangular lines in Cd as H.v/ for some order d complex Hadamard matrix H and constant v 2 C, and furthermore that these are the only dimensions for which this is possible. We ﬁrstly give examples of the construction in each of these three dimensions. Although the examples in dimensions 2 and 3 coincide with examples previously found using another construction method [19, Section 3.4], we include them here to illustrate a new connection between the three dimensions via a common construction. Example 2.2. Let H be the following order 2 complex Hadamard matrix: Â Ã 1 i H D : 1 i

Then H.v/ consists of the following four vectors

.v i / .v i /

.1 vi / .1 vi/ p C2 v 1 .1 3/.1 / which are equiangular in for D 2 C C i . Example 2.3. Let H be the following order 3 complex Hadamard matrix: 0 1 11 1 H D @ 1!!2 A !1!2 A Simple Construction of Complex Equiangular Lines 163

Then H.v/ consists of the following nine vectors

.v 1 1 / .v ! !2 / .v! 1 !2 /

.1 v 1 / .1 v!!2 / .! v !2 /

.1 1 v / .1 ! v!2/ .! 1 v!2/ which are equiangular in C3 for v D2. Example 2.4. Let H be the following order 8 Hadamard matrix: 0 1 11111111 B C B 1 1111111 C B C B 111 1111 1 C B C B 1 1 1111 11C B C : H D B 11111 1 1 1 C B C B C B 1 111 1111C @ 111 1 1 111A 1 1 111111

Then H.v/ consists of the following 64 vectors:

.v1111111/.1v111111/ .v 1111111/ . 1 v111111/ .v 1 1 1111 1/ . 1 v 1 1111 1/ .v 1 1111 11/.1v 1111 11/ .v 1 1 1 1 1 1 1/ . 1 v 1 1 1 1 1 1/ .v 111 1111/.1v11 1111/ .v 1 1 1 1 111/.1v1 1 1 111/ .v 1 111111/ . 1 v 111111/ 164 J. Jedwab and A. Wiebe

.11v11111/.111v1111/ .1 1v11111/ . 1 11v1111/ .1 1 v 1111 1/ . 1 1 1 v111 1/ .1 1 v111 11/.11 1v11 11/ .1 1 v 1 1 1 1 1/ . 1 1 1 v 1 1 1 1/ .1 1v1 1111/.111v 1111/ .1 1 v 1 1 111/.111 v 1 111/ .1 1 v11111/ . 1 1 1v1111/

.1111v111/.11111v11/ .1 111v111/ . 1 1111v11/ .1 1 1 1v11 1/ . 1 1 1 11v1 1/ .1 1 11v1 11/.11 111v 11/ .1 1 1 1 v 1 1 1/ . 1 1 1 1 1 v 1 1/ .1 111 v111/.1111 1v11/ .1 1 1 1 v 111/.111 1 1 v11/ .1 1 11v111/ . 1 1 111v11/

.111111v1/.1111111v/ .1 11111v1/ . 1 111111v/ .1 1 1 111v 1/ . 1 1 1 1111 v/ .1 1 1111 v1/.11 1111 1v/ .1 1 1 1 1 1 v 1/ . 1 1 1 1 1 1 1 v/ .1 111 11v1/.1111 111v/ .1 1 1 1 1 1v1/.111 1 1 11v/ .1 1 1111v1/ . 1 1 11111v/ which are equiangular in C8 for v D1 C 2i. It is easily veriﬁed by hand that each of the sets of vectors in Examples 2.2, 2.3 and 2.4 comprises a set of d2 equiangular lines in their respective dimensions.

3 Allowable Construction Parameters

The main theorem of this paper is the following, in which we characterize all dimensions d,orderd complex Hadamard matrices H and constants v 2 C for which one can construct d2 equiangular lines as H.v/. Theorem 3.1. Let d 2. Let H be an order d complex Hadamard matrix and v 2 C be a constant. Then H.v/ is a set of d2 equiangular lines if and only if one of the following holds: A Simple Construction of Complex Equiangular Lines 165 n 1 p 1 p 1 p 1. d D 2 and v 2 2 .1 ˙ 3/.1 C i/; 2 .1 ˙ 3/.1 i/; 2 .1 ˙ 3/.1 C i/, p o 1 .1 3/.1 / ; 2 ˙ i p 2. d D 3 and v 2f0; 2;1 ˙ 3ig; 3. d D 8 and H is equivalent to a real Hadamard matrix and v 2f1 ˙ 2ig.

Notice that if H.v/ is a set of equiangular lines then, for any complex Hadamard matrix H0 that is equivalent to H,thesetH0.v/ is also a set of equiangular lines. This is because permutation of the rows of H does not change the set H.v/; permutation of the columns of H applies a ﬁxed permutation to the coordinates of all vectors of H.v/; multiplication of a row of H by a constant c 2 C of magnitude 1 multiplies one vector in each set Hj.v/ by c; and multiplication of a column of H by a constant c 2 C of magnitude 1 multiplies a ﬁxed coordinate of each vector of H.v/ by c.In each case, the magnitude of the inner product between pairs of distinct vectors in H.v/ is unchanged. There are only three types of inner product that can arise between distinct vectors of H.v/:

(i) the inner product of two distinct vectors within a set Hj.v/, (ii) the inner product of two vectors of distinct sets Hj.v/; Hk.v/ which are derived from the same row of H, (iii) the inner product of two vectors of distinct sets Hj.v/; Hk.v/ which are derived from distinct rows of H. Therefore H.v/ is a set of d2 equiangular lines if and only if the equations obtained by equating the magnitudes of every inner product of type (i), (ii) and (iii) have a solution. In Lemma 3.1 we show that only one magnitude occurs for all inner products of type (i) and likewise for all inner products of type (ii). In Propositions 3.1 and 3.2 and Theorem 3.2 we show that, for dimensions 2, 3 and 8, inner products of type (iii) take values in only a small set. It is then straightforward to characterize the solutions obtained by equating magnitudes, for these three dimensions, and to show that the corresponding equations have no solutions in other dimensions. This establishes Theorem 3.1. Lemma 3.1. Let v D a C ib for a; b 2 R. For all d, every inner product of type (i) has magnitude ja2 C b2 1j and every inner product of type (ii) has magnitude j2a C d 2j. Proof. Two vectors having inner product of type (i) are derived from distinct rows of H; thus, their inner product in H is 0. In H.a C ib/ these vectors are multiplied in the same coordinate by a C ib, giving an inner product of magnitude ja2 C b2 1j. Two vectors having inner product of type (ii) are derived from the same row of H; thus, their inner product in H is d (given by a contribution of 1 from each coordinate of the vectors). In H.a C ib/ these vectors are multiplied in different coordinates by a C ib, giving an inner product of magnitude j2a C d 2j. 166 J. Jedwab and A. Wiebe

Proposition 3.1 (d D 2). Let H be an order 2 complex Hadamard matrix. Then n p H.v/ is a set of 4 equiangular lines in C2 if and only if v 2 1 .1 ˙ 3/.1 C i/, o 2 1 p 1 p 1 p 2 .1 ˙ 3/.1 i/; 2 .1 ˙ 3/.1 C i/; 2 .1 ˙ 3/.1 i/ . Proof. Up to equivalence, the only order 2 complex Hadamard matrix [12, Prop. 2.1] is Â Ã 11 H D : 1 1

Both inner products of type (iii) that occur in H.a C ib/ (where a; b 2 R)have magnitude j.a C ib/ .a C ib/jDj2bj. Using Lemma 3.1, H.a C ib/ is therefore a set of equiangular lines if and only if we can solve the equations

ja2 C b2 1jDj2ajDj2bj for a; b 2 R: n o 1 p 1 p This can be done exactly when a 2 2 .1 ˙ 3/; 2 .1 ˙ 3/ and b D˙a. 3 3 Proposition 3.2 (d D ). Let H be an order complex Hadamard matrix.p Then H.v/ is a set of nine equiangular lines in C3 if and only if v 2f0; 2;1 ˙ 3ig. Proof. Let ! D e2i=3. Up to equivalence, the only order 3 complex Hadamard matrix [12, Prop. 2.1] is 0 1 11 1 H D @ 1!!2 A : 1!2 !

All inner products of type (iii) that occur in H.v/ are derived from rows of H having inner product 1 C ! C !2 D 0.InH.v/, each of these inner products takes the form !j.v v! !2/ !j.v ! v!2/ 0; 1; 2 v C C or C C for some j 2f pg.For D a C ibpwith a; b 2 R, these inner products have magnitude ja 1 C b 3j and ja 1 b 3j, respectively, and both magnitudes occur. Using Lemma 3.1, H.a C ib/ is therefore a set of equiangular lines if and only if we can solve the equations p p ja2 C b2 1jDj2a C 1jDja 1 C b 3jDja 1 b 3j for a; b 2 R: p This can be done exactly when .a; b/ 2f.0; 0/; .2;0/, .1; ˙ 3/g. We now complete the proof of our main result, by showing that if H.v/ is a set of d2 equiangular lines for d >3then we must be in case (3) of Theorem 3.1. A Simple Construction of Complex Equiangular Lines 167

Theorem 3.2. Let d >3and let H be an order d complex Hadamard matrix. Then H.v/ is a set of d2 equiangular lines if and only if d D 8 and H is equivalent to a real Hadamard matrix and v 2f1 ˙ 2ig.

Proof. Let H D .hjk/ be an order d complex Hadamard matrix. We consider two cases. Case 1 is where, for every pair of distinct rows of H, all summands of the inner product of the rows take values in a set f;g for some 2 C (depending on the row pair) of magnitude 1. We now show that, in this case, H is equivalent to some real Hadamard matrix H0. Firstly transform each entry of the ﬁrst row of H to be 1 C 1, by multiplying each column k of H by the constant h1k 2 of magnitude 1. Then all summands of the inner product of the resulting rows 1 and j take values in a set fj; jg for some j 2 C of magnitude 1, and so all entries of row j lie in ; 1 C f j jg. Multiply each row j by the constant j 2 of magnitude 1 to obtain a real Hadamard matrix H0. We next show that all inner products of type (iii) that occur in H0.v/ have one of exactly two magnitudes. All such inner products are derived from rows of H0 having d .1/ d . 1/ d 2 >3 0.v/ inner product 2 C 2 ,and 2 since d .InH , each of these inner . 1/j v v . d 2/.1/ d . 1/ . 1/j.v v 2/ products takes the form C C 2 C 2 D C j d d j or .1/ v v C . 2 1/.1/ C . 2 1/.1/ D .1/ .v v/ for some j 2f0; 1g. For v D a C ib with a; b 2 R, these inner products have magnitude j2a 2j and j2bj, respectively, and both magnitudes occur. Using Lemma 3.1, H.a C ib/ is therefore a set of equiangular lines if and only if we can solve the equations

ja2 C b2 1jDj2a C d 2jDj2a 2jDj2bj for a; b 2 R:

This can be done exactly when d D 8 and .a; b/ 2f.1; ˙2/g. Case 2 is where the summands of the inner product of some pair of distinct rows of H are 1, 2, 3 for distinct j 2 C of magnitude 1, together with 1;:::;d3 for d 3>0other elements j 2 C of magnitude 1 (not necessarily distinct from each other or from the j). Thus in H.v/ there are three pairs of vectors, derived from this pair of rows, having inner products .v 1/1 C .v 1/1, .v 1/1 C .v 1/2 and .v 1/1 C .v 1/3. We now show that there is no a; b 2 R for which H.aCib/ is a set of equiangular lines. Suppose otherwise, for a contradiction. Then the above three inner products have equal magnitude for v D a C ib,sothat

j.a 1 C ib/1 C .a 1 ib/1jDj.a 1 C ib/1 C .a 1 ib/2j

Dj.a 1 C ib/1 C .a 1 ib/3j: (2)

Notice that from Lemma 3.1, we cannot havep.a; b/ D .1; 0/ as this would imply 2 2 d D 0. Therefore .a 1 C ib/1 ¤ 0 and .a 1/ C b ¤ 0.Nowthej are all distinct with magnitude 1,so.a 1 ib/1, .a 1 ib/2, .a 1 ib/3 are 168 J. Jedwab and A. Wiebe p all distinct with magnitude .a 1/2 C b2 >0. But then it is clear geometrically that only two of these three quantities can have equal magnitude after being added to .a 1 C ib/1 ¤ 0. This contradicts (2).

Acknowledgements We thank Matt DeVos for his interest in this construction and the resulting helpful discussion and important insight regarding the proof of Theorem 3.2. We are grateful to Huangjun Zhu for his generosity in pointing out the unitary transformation involving (1). J. Jedwab is supported by an NSERC Discovery Grant. A. Wiebe was supported by an NSERC Canada Graduate Scholarship.

References

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Ilias S. Kotsireas, Jennifer Seberry, and Yustina S. Suharini

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract Algorithms to find new orders of skew-Hadamard matrices by complete searches are not efficient and require extensive CPU time. We consider a method relying on pre-calculation of inner product vectors aiming to reduce the search space. We apply our method to the algorithm of Seberry–Williamson to construct skew-Hadamard matrices. We find all possible solutions for Ä 29.Weusethese results to improve analysis in order to reduce the search space.

Keywords Hadamard matrices • Seberry-Williamson array • Skew-Hadamard matrices • Good matrices • Supplementary difference sets • 05B20

1 Introduction

1.1 Deﬁnitions

A Hadamard matrix, H,ofordern is a square ˙1 matrix whose rows (and columns) > are pairwise orthogonal, that is HH D nIn. Hadamard matrices of order n are

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. I.S. Kotsireas () Department of Physics & Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5 e-mail: [email protected] J. Seberry Centre for Computer and Information Security Research, SCSSE, University of Wollongong, Wollongong, NSW 2522, Australia e-mail: [email protected] Y.S. Suharini Department of Informatics Engineering, Institute of Technology Indonesia, Tangerang, Banten, Indonesia e-mail: [email protected]

© Springer International Publishing Switzerland 2015 171 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_14 172 I.S. Kotsireas et al. conjectured to exist for all orders n Á 0.mod 4/. A weighing matrix, W D W.n; k/, > of order n and weight k has elements 0; ˙1 and satisﬁes WW D kIn.IfaHadamard matrix M, can be written in the form M D I C S where S> DS,thenM is said to be a skew-Hadamard matrix. Skew Hadamard matrices are also conjectured to exist for all orders n Á 0.mod 4/. However, compared with the knowledge regarding the existence of Hadamard matrices very little is known regarding the existence of skew-Hadamard matrices. Goethals and Seidel discovered a skew-Hadamard matrix of order 36 in 1970 [8]. Seberry discovered a skew-Hadamard matrix of order 92 in 1971 [16]. In a series of papers [3, 5–7] -Dokovic´ discovered skew-Hadamard matrices of several new orders. In all our examples minus (“”) is used to denote minus one (“1”). Example 1.1 (Hadamard Matrices). 2 3 2 3 1111 1 11 Ä 6 7 6 7 11 6 117 6 1 7 H2 D H4symmetric D 4 5 H4skew-type D 4 5 1 1 1 1 1 1 111

1.2 Circulant and Type 1 Matrix Basics

Because it is so important to the rest of our work we now spend a little effort to establish why the properties required for Williamson matrices are so important. We deﬁne the shift matrix, T of order t by 2 3 010 6 7 6 000 7 T D 6 : : 7 : 4 : : 5 100

So any circulant matrix, of order t and ﬁrst row x1; x2; ; xt,thatis, 2 3 x1 x2 x3 xt 6 7 6 xt x1 x2 xt1 7 6 7 6 xt1 xt x1 xt2 7 (1) 6 : : 7 4 : : 5 x2 x3 x4 x1 can be written as the polynomial

> 2 t1 x1T C x2T C x3T xtT : Inner Product Vectors for Skew-Hadamard Matrices 173

We now note that polynomials commute, so any circulant matrices of the same order t commute. We deﬁne the back-diagonal matrix, R of order t by 2 3 001 6 7 6 000 7 R D 6 : : 7 : 4 : : 5 100

We note TaR is a polynomial,for integer a, so that, similarly, any back-circulant matrix, of order t and ﬁrst row x1; x2; ; xt,thatis, 2 3 x1 x2 x3 xt 6 7 6 x2 x3 x4 x1 7 6 7 6 x3 x4 x5 x2 7 6 : : 7 4 : : 5

xt x1 x2 xt1 can be written as the polynomial

> 2 t1 x1T R C x2TR C x3T R xtT R:

We now note that polynomials commute, so any two back-circulant matrices of thesameordert commute.

Deﬁnition 1.1. A circulant matrix C D .cij/ of order t is a matrix which satisﬁes the condition that

cij D c1;jiC1 D ciCk;jCk (2) where j i C 1 is reduced modulo t [17]. A back circulant matrix B D .bij/ order n is a matrix with property that

bij D b1;iCj1 D biCk;jk (3) where i C j 1 is reduced modulo t [17]. Two matrices, X and Y ofthesameordert which satisfy

XY> D YX> (4) are said to be amicable matrices. A back circulant matrix has transpose as the same as itself, so it is also a symmetric matrix. 174 I.S. Kotsireas et al.

Lemma 1.1. If X is a back circulant matrix and Y is a circulant matrix, then X and Y are amicable matrices because XY> D YX>,see[17]. Here are examples of amicable matrices of order 3.(1,1)-matrices are used rather than other matrices because we are talking about Hadamard matrices whose elements only ones and minus ones. 2 3 2 3 2 3 1 1 1 113 C D 4 11 5 , B D 4 1 5 , CB> D BC> D 4 1 315 11 1 31 1

In all our definitions of circulant and back-circulant matrices we have assumed that the rows and columns have been indexed by the order, that is for order t,the rows are 1, 2, ; t and similarly for the columns. The internal entries are then defined by the first row using a 1:1 and onto mapping. However we could have indexed the rows and columns using the elements of a group G, with elements g1, g2,,gt. Loosely a type one matrix will then be defined so the .ij/ element depends on a 1:1 and onto mapping of gj gi for type 1 matrices and on gj C gi for type 2 matrices. We use additive notation, but that is not necessary. Wallis and Whiteman [18] have shown that circulant and type 1 can be used interchangeably and can the terms back-circulant and type 2. This can be used to explore similar theorems in more structured groups.

Deﬁnition 1.2 (Additive Property). k circulant matrices, A1, A2, Ak,ofordert with elements ˙1 only and which satisfy

Xk > ; AIAi D ktIt iD1 will be said to satisfy the additive property (for k).

1.3 Historical Background

Hadamard matrices ﬁrstly we appeared in the literature in an 1867 paper written by Sylvester [14]. In 1893 Hadamard matrices appear, called matrices on the unit circle, they satisﬁed Hadamard’s inequality for the determinant of matrices with entries within the unit circle [9]. Many matrices were found by Scarpis in 1898 [12]. In 1933 Paley [11] conjectured that the matrices existed for all positive integer orders divisible by 4. This has become known as the Hadamard conjecture: Conjecture 1.1. Hadamard matrices exist for all orders 1, 2, 4t,wheret is a positive integer. Inner Product Vectors for Skew-Hadamard Matrices 175

After Paley’s work [11] the following orders less than 200: 23(4), 29(4), 39(3), 43(3), 47(4), 59(12), 65(3), 67(5), 73(7), 81(3), 89(4), 93(3), 101(10), 103(3), 107(10), 109(9), 113(8), 119 for p(3), 127(25), 133(3), 149(4), 151(5), 153(3), 163(3), 167(4), 179(8), 183(3), 189(3), 191, 193(3) were unresolved. The number in brackets, if it is provided, indicates that one of order 2at is given in Seberry and Yamada [13]. The ﬁrst unsolved cases are currently for primes p D 167, 179, 191 and 193.

2 Williamson Array

In 1944 [19] Williamson proposed using what has come known as the Williamson array. It can be shown, for example see [17], that if we can calculate suitable matrices, of order t, satisfying the additive property for 4, they can be plugged-in to his array to give Hadamard matrices of order 4t. Hence Deﬁnition 2.1 (Williamson Matrices). Four circulant symmetric matrices, A, B, C and D,ofordert with elements ˙1 only and which satisfy

> > > > AA C BB C CC C DD D 4tIt; will be called Williamson matrices of order t. Theorem 2.1 (Williamson’s Theorem). Suppose A, B, C and D of order t are Williamson matrices. Then these matrices can be plugged into the Williamson array 2 3 ABCD 6 7 6 BADC7 Warray D 4 5 (5) CDAB D CBA to obtain a Hadamard matrix of order 4t. > Remark 2.1. Crucial part of proof. When we look at the terms of WarrayWarray for, say the (2,3) element we obtain

BC> C AD> DA> C CB> but for Williamson matrices A D A>, B D B>, C D C> and D D D>.Furthermore A, B, C and D pairwise commute so

BC> C AD> DA> C CB> DBC C AD DA CCB DBC C BC DA C DA D 0;

> formally for all off diagonal elements of WarrayWarray. 176 I.S. Kotsireas et al.

Fig. 1 Williamson matrix of ⎡ ⎤ 1 −− 1 −− 1 −− 111 order 4 3 ⎢ − 1 − − 1 − − 1 − 111⎥ ⎢ ⎥ ⎢ −− 1 −− 1 −− 1 111⎥ ⎢ ⎥ ⎢ − 111 −−−−− 1 −−⎥ ⎢ ⎥ ⎢ 1 − 1 − 1 − −−−− 1 − ⎥ ⎢ ⎥ ⎢ 11− −− 1 −−−−− 1 ⎥ W = ⎢ ⎥ 12Williamson ⎢ − 111111 −− 1 −−⎥ ⎢ ⎥ ⎢ 1 − 1 111− 1 − − 1 − ⎥ ⎢ ⎥ ⎢ 11− 111−− 1 −− 1 ⎥ ⎢ ⎥ ⎢ −−−− 11− 111 −−⎥ ⎣ −−− 1 − 1 1 − 1 − 1 − ⎦ −−− 11− 11− −− 1

Example 2.1. A Williamson matrix of order 4 3 is shown in Fig. 1. Many researchers have continued to search for Williamson and Williamson- like matrices (Williamson type, good matrices, best matrices, Goethals–Seidel type matrices, see below) which satisfy the additive property for 4.

2.1 Existence of Williamson Matrices

In the 1960s Hadamard matrices were studied as part of a program to find the best possible error correction codes to be used to transmit data from deep space back to Earth. Williamson did not use a computer to construct Williamson matrices but used some clever number theory. After Williamson the following orders t: 23, 29, 47, 59, 65, 67, 77, 103, 105, 107, 111, 119, 133, 143, 151, 155, 161, 163, 167, 171, 179, 183, 185, 191, 203, 207, 209, 215, 219, 221, 223, 227, 237, 245, 247, 249, 251, 253, 259, 267, 273, 283, 287, 291, 299 less than 300 were unknown. Baumert et al. [2] used a computer with the Williamson construction to construct the order 92 D 4 23. More recent results have been sporadic or the results of extensive calculations. Baumert and Hall [1] gave a very pretty construction to find the Hadamard matrix of order 156 which used what came to be called Baumert-Hall arrays. Now new methods were being discovered to find Hadamard matrices and some of these used skew-Hadamard matrices. In [4] -Dokovic´ showed that there is no Williamson matrix for t D 35.The computational state-of-the-art paper on Williamson matrices is [10], in which the authors show that Williamson matrices do not exist for t D 47; 53; 59. Inner Product Vectors for Skew-Hadamard Matrices 177

2.2 Seberry–Williamson Arrays

Williamson matrices are symmetric: the Hadamard matrix they form is neither symmetric nor skew-symmetric. As interest rose in the usefulness of skew-Hadamard matrices for further construction modifications of the Williamson array were proposed to enable computer search. This led to what we will now call the Seberry– Williamson array, which first appeared in the Ph.D. thesis of (Seberry) Wallis) [15]. Definition 2.2 (Good Matrices). Four circulant matrices, of order t with elements ˙1 only, where A, is skew-symmetric (.A I/> D.A I/)andB, C and D,and which satisfy

> > > > AA C BB C CC C DD D 4tIt; will be called good matrices of order t. Theorem 2.2 ((Seberry) Wallis’s Theorem). Suppose A, B, C and D of order t are good matrices. Then these matrices can be plugged into the Seberry–Williamson array 2 3 ABRCRDR 6 7 6 BR A DR CR 7 SWarray D 4 5 (6) CR DR A BR DR CR BR A to obtain a skew Hadamard matrix of order 4t. Remark 2.2. The proof is similar to that for the Williamson array after noting that circulant matrices and back-circulant matrices are amicable which pairs of back- circulant matrices commute. These were some of the first searched for by computer. The ones given in (Seberry) Wallis’ Ph.D. thesis [15] were found on a PDP6 taking over 100 h per week for many months in 1969. The limitation was basically the RAM memory of 4K. The matrices for 92 D 4 23 [16] were found at the University of Newcastle, NSW in 1970 using about 200 h of CPU time. Again RAM was the largest constraint. Example 2.2 (Seberry–Williamson Matrix of Order 43). The Seberry–Williamson matrix for first rows A D 11; B D 1 ; C D 1 ; D D 111 is shown first in Fig. 2. Because of the polynomial nature of back-circulants we could have also said that the WSeberry–Williamson skew-Hadamard matrix just described we could have used the equivalent matrix WSW shown in Fig. 2. 178 I.S. Kotsireas et al.

⎡ ⎤ ⎡ ⎤ 11− −− 1 −− 1 111 11− 1 −− 1 −− 111 ⎢ − 11− 1 − − 1 − 111⎥ ⎢ − 11−− 1 −− 1 111⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 − 1 1 −− 1 −− 111⎥ ⎢ 1 − 1 − 1 − − 1 − 111⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 11− 11− 11111− ⎥ ⎢ − 1111− 111− 11⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 − 1 − 111111 − 1 ⎥ ⎢ 11− − 1111111− ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − 111 − 1 111− 11⎥ ⎢ 1 − 1 1 − 1 1111 − 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 11− −−− 11− −− 1 ⎥ ⎢ − 11−−− 11− 1 −−⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 − 1 −−−− 11− 1 − ⎥ ⎢ 11− −−−− 11−− 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − 11−−− 1 − 1 11− ⎥ ⎢ 1 − 1 −−− 1 − 1 − 1 − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −−−−− 1 11− 11− ⎥ ⎢ −−− 1 −−− 1111− ⎥ ⎣ −−−− 1 − 1 − 1 − 11⎦ ⎣ −−−−− 1 11− − 11⎦ −−− 1 −−− 111 − 1 −−−− 1 − 1 − 1 1 − 1

Fig. 2 WSeberry–Williamson and WSW

3 IPV Vectors

Deﬁnition 3.1. The inner product of rows i and j of any matrix, G D .gxy/,of order t is Xt gikgjk: kD1

Example 3.1 (Inner Products of Matrix G1). 2 3 11 1 6 7 6 11 1 7 6 7 6 1 11 7 : G1 D 6 7 4 1 115 1 1 1

Inner product between the ﬁrst row and the second row is calculated as X5 .g1k g2k/ D .1 1/ C .1 1/ C .1 1/ C .1 1/ C .1 1/ D3: kD1

Lemma 3.1 (Mirror Lemma). Consider the inner products of rows of a circulant (back-circulant) matrices of order t. The inner product for the jth and `th row is the same as the inner product for the ﬁrst and ` jth the rows so that there are at most t1 2 distinct inner products. Inner Product Vectors for Skew-Hadamard Matrices 179

Deﬁnition 3.2. Let G D .gxy/ be a circulant (back-circulant) matrix of order t. tC1 We call the 1 vector with entries .p2; p3; ; p tC1 /, that is the vector has 2 2 tC1 coordinates the inner products of rows 1 and 2, 1 and 3, ,1and 2 , that is the vector Xt . / IPVxij D x1;kx1;kjCi kD1 the inner product vector.

Remark 3.1. A naiveÂ Ã approach to ﬁnding the inner product vectors of any n n n 2 matrix would take 2 n calculations. Using the mirror lemma we have reduced the number of by 2. Remark 3.2. Each row of a circulant ˙1 matrix can be considered as an integer, uniquely, by replacing the elements 1 by zero and converting the sequence to decimal. Thus a circulant matrix of order t can be represented by an integer of size the least integer greater than ln2t. This means any sequence we would consider in search for skew-Hadamard matrices using current technology can be represented by one word of space. This is used in the results section to describe the solution matrices. Lemma 3.2. We now consider circulant (or back circulant) matrices of order t, t odd, with entries ˙1 only. Then • if t Á 1.mod /4, then the entries of the IPV are Á 1.mod /4; • if t Á 3.mod /4, then the entries of the IPV are Á 3.mod /4; • the coordinates in an IPV are integers between .t 4/ and t; • the sum of the coordinates in an IPV is 0.

3.1 Sums of Squares of First Rows of Williamson Matrices

We notice that for arrays which have ˙1 matrices, A, B, C, D of order t, satisfying the 4-additive property, plugged into them where

> > > > AA C BB C CC C DD D 4tIt:

Then if e is the 1 m matrix of all ones and the row sums of A, B, C, D are a, b, c, and d, respectively. Then

eA D ae; eB D be; eC D ce; eD D ad; and

> > > > 2 2 2 2 e.AA C BB C CC C DD / D 4mIm D a e C b e C c e C d e D 4me: 180 I.S. Kotsireas et al.

3.1.1 Sums of Squares of First Rows of Good Matrices for Seberry–Williamson Matrices

Let A, B, C and D be good matrices of order t with ﬁrst row sums a, b, c and d.Then using exactly the same proof as for the ﬁrst rows of Williamson matrices, except that the row-sum of the skew-type matrix is a DC1 we have for four good matrices of order t

4t D 1 C b2 C c2 C d2:

In the case of the skew-Williamson array a is always DC1.

4 Computational Results on IPV for t D 3; 5; 7;9; 11; 13

A naive algorithm to ﬁnd matrices for the skew-array or good matrices was used on various machines to obtain comparison run times for t D 1; 3; 5; ; 27; 29; 31. These show that more sophistication is needed to make the results required attainable for larger t. The results obtained for t D 1; ;13are given in the last section. Further results are available from the authors for t D 15; 17; 19; 21; 23; 25; 27; 29. The computations were performed on Shared Hierarchical Academic Research Computing Network (SHARCNET) and RQCHP clusters. This work was made possible by the facilities of the SHARCNET (www.sharcnet.ca) and Compute/Calcul Canada. Computations were made on the Mammouth supercomputer managed by Calcul Québec and Compute Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), NanoQuébec, RMGA and the Fonds de recherche du Québec—Nature et technologies (FRQ-NT). We consider the inner product vectors IPVs for the Seberry–Williamson construction for skew-Hadamard matrices which uses the ﬁrst rows of the corresponding good matrices: A (skew-type), B (symmetric), C (symmetric) and D (symmetric) in their construction. We give fa1; a2; ; a 1 g, 2 .t1/ fb1; b2; ; b 1 g, fc1; c2; ; c 1 g,andfd1; d2; ; d 1 g for the 2 .t1/ 2 .t1/ 2 .t1/ coordinates of the IPVA, IPVB, IPVC and IPVD, respectively. In each case we note that for the coordinates in the IPV 1 •wehaveai; bi; ci; di, 2 Ä i Ä 2 .t 1/ always takes only integer values Á t .mod 4/, see Lemma 3.2. 1 • summing ai C bi C ci C di for each i D 1; 2; ; 2 .t 1/ always gives 0 (this is required for skew-Williamson matrices); •wehave4t D .1/2 C .˙b/2 C .˙c/2 C .˙d/2 in every case (this is required for skew-Williamson matrices). Inner Product Vectors for Skew-Hadamard Matrices 181

4.1 The IPVs for t=3

There is only one solution from a complete search for t D 3. It involves four matrices A, B, C,andD which have row sums

.1/.1/.1/.3/:

1. the maximum coordinate that appears is C3 and the absolute minimum coordinate that appears is 1; 2. the sum of the row sums is .1/ C .1/ C .1/ C 3 D 0; 3. 4t D 12 D .1/2 C .1/2 C .1/2 C .3/2; 4. the ﬁrst rows of the corresponding good matrices A (skew-type), B (symmetric), C (symmetric) and D (symmetric) are 1 1 -1; 1 -1 -1; 1 -1 -1; 1, 1, 1; 5. the integers which correspond to these ﬁrst rows are 6, 4, 4, 7;

4.2 The IPVs for t=5

A complete search yields two possible solutions for t D 5. They each involve four good matrices A (skew-type), B (symmetric), C (symmetric) and D (symmetric) which have row sums

5:1 W .1; 3/.3; 1/.1; 1/.1; 1/ and

5:2 W .3; 1/.1; 3/.1; 1/.1; 1/

1. the maximum absolute value coordinate that appears in any of the IPVs is 1 and the minimum absolute value coordinate that appears in any of the IPVs is 3; 2. summing ai C bi C ci C di for i=1,2,3,4,fortheIPVsgives.2/ C .2/ C .2/ C .2/ D 0 in both cases; 3. 4t D 20 D .1/2 C .1/2 C .3/2 C .3/2; 4. the ﬁrst rows of the corresponding good matrices A (skew-type), B (symmetric), C (symmetric) and D (symmetric) are 182 I.S. Kotsireas et al.

1 1 1 -1 -1; 1 1 -1 -1 1; 1 -1 -1 -1 -1; 1,-1,-1 -1 -1; and 1 1 -1 1 -1; 1 -1 1 1 -1; 1 -1 -1 -1 -1; 1,-1,-1 -1 -1; 5. the integers which correspond to these ﬁrst rows are 30, 25, 16 and 16; and 26, 22, 16 and 16;

4.3 The IPVs for t=7

A complete search gives a total of seven solutions. 1. the IPV vectors obtained are: 7.1 :(-5, 3, -1) (-1, -5, 3) (3, -1, -5) (3, 3, 3) 7.2 :(-5, 3, -1) (-1, -1, 3) (3, -1, -1) (3, -1, -1) 7.3 :(-1, -5, 3) (-5, 3, -1) (3, -1, -5) (3, 3, 3) 7.4 :(-1, -5, 3) (-1, 3, -1) (-1, 3, -1) (3, -1, -1) 7.5 :(-1, -1, -1) (-1, -1, 3) (-1, 3, -1) (3, -1, -1) 7.6 :(3, -1, -5) (-5, 3, -1) (-1, -5, 3) (3, 3, 3) 7.7 :(3, -1, -5) (-1, -1, 3) (-1, -1, 3) (-1, 3, -1) 2. the IPV values range from 5 to +3. All coordinates n the IPV are Á 3.mod 4/; 3. the sum of the row sums of the IPV is .3/ C .3/ C .3/ C .9/ D 0 in three cases and .3/ C .1/ C .1/ C .1/ D 0, in four cases; 4. 4t D 28 D .1/2 C .1/2 C .1/2 C .5/2 D .1/2 C .3/2 C .3/2 C .3/2; 5. the integers which correspond to the ﬁrst rows to make the good matrices A (skew-type), B (symmetric), C (symmetric) and D (symmetric) are: 106 76 97 64 106 109 115 115 89 82 97 64 89 94 94 115 75 109 94 115 120 82 76 64 120 109 109 94

4.4 The IPVs for t=9

A complete search gives a total of seven solutions. 9.1 :(-3, 1, -3, 1) (-3, -3, 1, 1) (5, 1, -3, -3) (1, 1, 5, 1) 9.2 :(1, -3, -3, 1) (1, -3, 1, -3) (-3, 5, -3, 1) (1, 1, 5, 1) 9.3 :(1, 1, -3, -3) (-3, 1, 1, -3) (1, -3, -3, 5) (1, 1, 5, 1) 1. The IPVs obtained are shown above. 2. the IPV values range from 3 to +5. All coordinates in the IPV are Á 1.mod 4/; Inner Product Vectors for Skew-Hadamard Matrices 183

3. the sum of the row sums of the IPV is .4/ C .4/ C .0/ C .0/ D 0 in all three cases; 4. 4t D 36 D .1/2 C .1/2 C .3/2 C .5/2; 5. the integers which correspond to the ﬁrst rows to make the good matrices A (skew-type), B (symmetric), C (symmetric) and D (symmetric) are: 468 358 385 475 369 316 322 475 279 421 280 475 In future research we will seek to ﬁnd the relationships between the IPVs to limit our search space and time.

4.5 The IPVs for t=11

A complete search gives a total of 15 solutions. 11.1:(-5,-1,3,-5,3)(-1,-1,-5,3,3)(7,3,-1,-5,-5)(-1,-1,3,7,-1) 11.2:(-5,-1,3,-1,-1)(-5,3,-1,-1,3)(3,-5,-1,3,-1)(7,3,-1,-1,-1) 11.3:(-5,-1,3,-1,-1)(3,-5,-5,-1,7)(3,-1,3,-1,-5)(-1,7,-1,3,-1) 11.4:(-5,3,-5,3,-1)(-5,-1,7,-5,3)(3,-5,-1,3,-1)(7,3,-1,-1,-1) 11.5:(-1,-5,-1,-1,3)(-5,3,-1,-1,3)(7,3,-1,-5,-5)(-1,-1,3,7,-1) 11.6:(-1,-5,-1,-1,3)(-1,3,3,-5,-1)(3,-5,-1,3,-1)(-1,7,-1,3,-1) 11.7:(-1,-5,3,3,-5)(-1,3,3,-5,-1)(3,-5,-5,-1,7)(-1,7,-1,3,-1) 11.8:(-1,-1,-1,3,-5)(-5,-1,7,-5,3)(-1,-1,-5,3,3)(7,3,-1,-1,-1) 11.9:(-1,-1,-1,3,-5)(-5,3,-1,-1,3)(3,-1,3,-1,-5)(3,-1,-1,-1,7) 11.10:(-1,3,-5,-1,-1)(-1,-5,3,7,-5)(-1,3,3,-5,-1)(3,-1,-1,-1,7) 11.11:(-1,3,-5,-1,-1)(-1,-1,-5,3,3)(3,-1,3,-1,-5)(-1,-1,7,-1,3) 11.12:(3,-5,-5,-1,3)(-5,7,-5,3,-1)(3,-1,3,-1,-5)(-1,-1,7,-1,3) 11.13:(3,-1,-1,-5,-1)(-5,7,-5,3,-1)(3,-5,-1,3,-1)(-1,-1,7,-1,3) 11.14:(3,-1,-1,-5,-1)(-1,-1,-5,3,3)(-1,3,3,-5,-1)(-1,-1,3,7,-1) 11.15:(3,3,-1,-5,-5)(-5,3,-1,-1,3)(-1,-5,3,7,-5)(3,-1,-1,-1,7) 1. The IPVs obtained are shown above. 2. the IPV values range from 5 to +7. All coordinates in the IPV are Á 3.mod 4/; 3. the row sums of the IPV is .5/ C .1/ C .1/ C .7/ in all 15 cases; 4. 4t D 44 D .1/2 C .3/2 C .3/2 C .5/2; 5. the integers which correspond to the ﬁrst rows to make the good matrices A (skew-type), B (symmetric), C (symmetric) and D (symmetric) are: 1381 1486 1927 1096 1449 1717 1843 1537 1449 1657 1276 1282 1195 1462 1843 1537 1836 1717 1927 1096 1836 1867 1843 1282 1892 1867 1657 1282 1582 1462 1486 1537 184 I.S. Kotsireas et al.

1582 1717 1276 1072 1960 1741 1867 1072 1960 1486 1276 1156 1127 1402 1276 1156 1505 1402 1843 1156 1505 1486 1867 1096 1071 1717 1741 1072 Observations. 1. no solution for “A” contains a 7; 2. some “A” appear in more than one solution; 3. “C” is the only matrix for which the ﬁrst coordinate, here c1 D 7;

5 The IPVs for t=13

A complete search gave 36 solutions. The IPV values range from 7to+9.All coordinates in the IPV are Á 1.mod 4/;

(-7,1,1,-3,5,-3)(5,-3,-7,-3,1,1)(-3,1,1,5,-3,5)(5,1,5,1,-3,-3) (-7,1,1,1,-3,1)(-3,-7,1,5,1,-3)(1,1,-3,-3,5,5)(9,5,1,-3,-3,-3) (-7,1,1,1,-3,1)(-3,1,-3,-3,1,1)(5,-3,-3,1,5,1)(5,1,5,1,-3,-3) (-7,1,1,1,-3,1)(1,-7,-3,1,5,-3)(1,5,1,-3,-3,-7)(5,1,1,1,1,9) (-3,-7,5,1,-3,1)(1,5,1,-3,-3,-7)(-3,5,-3,1,1,5)(5,-3,-3,1,5,1) (-3,-3,-3,1,5,-3)(1,1,-7,-3,-3,5)(1,-3,5,5,-3,1)(1,5,5,-3,1,-3) (-3,-3,1,-3,-3,5)(-7,5,-3,1,1,-3)(5,-3,-3,1,5,1)(5,1,5,1,-3,-3) (-3,-3,5,-3,1,-3)(5,1,-3,1,-3,-7)(-3,1,1,5,-3,5)(1,1,-3,-3,5,5) (-3,1,-3,-3,1,1)(-3,-7,1,5,1,-3)(1,5,1,-3,-3,-7)(5,1,1,1,1,9) (-3,1,-3,-3,1,1)(-3,-3,1,1,-7,5)(5,1,-3,1,-3,-7)(1,1,5,1,9,1) (-3,1,-3,-3,1,1)(1,-7,-3,1,5,-3)(1,-3,5,-3,-7,1)(1,9,1,5,1,1) (-3,1,-3,5,-3,-3)(1,-3,5,-3,-7,1)(-3,5,1,-3,5,1)(5,-3,-3,1,5,1) (-3,1,1,1,-7,1)(-3,1,5,-7,1,-3)(5,-3,-7,-3,1,1)(1,1,1,9,5,1) (-3,1,1,1,-7,1)(1,-3,1,1,-3,-3)(-3,5,1,-3,5,1)(5,-3,-3,1,5,1) (-3,1,1,1,-7,1)(1,1,-7,-3,-3,5)(-3,1,5,-3,9,-3)(5,-3,1,5,1,-3) (-3,5,-7,1,-3,1)(-3,1,5,-7,1,-3)(1,-3,5,5,-3,1)(5,-3,-3,1,5,1) (1,-7,-3,1,1,1)(-7,1,-3,5,-3,1)(-3,1,5,-7,1,-3)(9,5,1,1,1,1) (1,-7,-3,1,1,1)(-3,-3,1,-7,5,1)(-3,9,-3,5,-3,1)(5,1,5,1,-3,-3) (1,-7,-3,1,1,1)(1,-3,1,1,-3,-3)(-3,5,-3,1,1,5)(1,5,5,-3,1,-3) (1,-3,-7,1,1,1)(-3,-3,1,1,-7,5)(1,5,1,-3,-3,-7)(1,1,5,1,9,1) (1,-3,-7,1,1,1)(-3,1,-3,-3,1,1)(1,-3,5,5,-3,1)(1,5,5,-3,1,-3) (1,-3,-7,1,1,1)(5,1,-3,1,-3,-7)(-3,-3,9,1,-3,5)(-3,5,1,-3,5,1) (1,-3,-3,-7,1,5)(-7,1,-3,5,-3,1)(1,5,5,-3,1,-3)(5,-3,1,5,1,-3) (1,-3,-3,5,1,-7)(-3,-3,1,1,-7,5)(1,1,-3,-3,5,5)(1,5,5,-3,1,-3) (1,-3,1,1,-3,-3)(-7,1,-3,5,-3,1)(-3,-3,1,-7,5,1)(9,5,1,1,1,1) (1,-3,1,1,-3,-3)(-7,5,-3,1,1,-3)(5,-3,-7,-3,1,1)(1,1,9,1,1,5) (1,-3,1,1,-3,-3)(-3,1,5,-7,1,-3)(1,1,-7,-3,-3,5)(1,1,1,9,5,1) (1,1,1,-7,1,-3)(-3,-3,1,1,-7,5)(1,-7,-3,1,5,-3)(1,9,1,5,1,1) (1,1,1,-7,1,-3)(-3,1,-3,-3,1,1)(-3,1,1,5,-3,5)(5,-3,1,5,1,-3) (1,1,1,-7,1,-3)(1,-3,5,-3,-7,1)(-3,5,-3,1,1,5)(1,-3,-3,9,5,-3) Inner Product Vectors for Skew-Hadamard Matrices 185

(1,1,1,-3,1,-7)(-7,1,-3,5,-3,1)(5,-3,-7,-3,1,1)(1,1,9,1,1,5) (1,1,1,-3,1,-7)(-7,5,-3,1,1,-3)(1,-3,5,5,-3,1)(5,-3,-3,-3,1,9) (1,1,1,-3,1,-7)(1,-3,1,1,-3,-3)(-3,1,1,5,-3,5)(1,1,-3,-3,5,5) (1,5,-3,-3,-3,-3)(-3,-3,1,-7,5,1)(-3,1,1,5,-3,5)(5,-3,1,5,1,-3) (5,-3,-3,-3,-3,1)(-3,-7,1,5,1,-3)(-3,5,-3,1,1,5)(1,5,5,-3,1,-3) (5,1,1,-3,-7,-3)(1,-7,-3,1,5,-3)(-3,1,1,5,-3,5)(-3,5,1,-3,5,1) 1. The IPVs obtained are shown above; 2. the row sums of the IPV are 6, 6, 6, 6. 3. 4t D 52 D .1/2 C .1/2 C .5/2 C .5/2 D .1/2 C .1/2 C .1/2 C .7/2; 4. the integers which correspond to the ﬁrst rows to make the good matrices A (skew-type), B (symmetric), C (symmetric) and D (symmetric) are: 6858 7267 5998 5116 6486 5734 6046 7951 6486 5362 6649 5116 6486 5020 7435 4192 5325 7435 6901 6649 4947 5902 7579 7831 4811 6805 6649 5116 4699 4600 5998 6046 5573 5734 7435 4192 5573 6757 4600 4240 5573 5020 7315 5122 5969 7315 7531 6649 5213 4852 7267 4360 5213 6925 7531 6649 5213 5902 5878 7783 4439 4852 7579 6649 6598 5530 4852 6145 6598 6505 5626 5116 6598 6925 6901 7831 7260 6757 7435 4240 7260 5362 7579 7831 7260 4600 7021 7531 7768 5530 7831 7783 7620 6757 6046 7831 7880 5530 6505 6145 7880 6805 7267 4612 7880 4852 5902 4360 7106 6757 5020 5122 7106 5362 5998 7783 7106 7315 6901 7069 6238 5530 7267 4612 6238 6805 7579 7411 6238 6925 5998 6046 8016 6505 5998 7783 186 I.S. Kotsireas et al.

4303 5734 6901 7831 6081 5020 5998 7531

6TheIPVsfort D 15; 17;19; 21; 23; 25; 27;29

We summarize the results for the above values of t below. Full details are available from the authors. • t D 15, a complete search gave 44 solutions. The values in the IPV range from 13 to C11. • t D 17, a complete search gave 16 solutions. The IPV values range from 11 to C9. • t D 19, a complete search gave 64 solutions. The IPV values range from 9 to C11. • t D 21, a complete search gave 60 solutions. The IPV values range from 11 to C9. • t D 23, a complete search gave 66 solutions. The values in the IPV range from 13 to C11. • t D 25, a complete search gave 90 solutions. The values in the IPV range from 15 to C13. • t D 27, a complete search gave 117 solutions. The values in the IPV range from 13 to C15. • t D 29, a complete search gave 71 solutions. The values in the IPV range from 15 to C11.

References

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9. Hadamard, J.: Resolution d’une question relat ive aux determinants. Bull. Sci. Math. 17, 240–246 (1893) 10. Holzmann, W.H., Kharaghani, H., Tayfeh-Rezaie, B.: Williamson matrices up to order 59. Des. Codes Crypt. 46(3), 343–352 (2008) 11. Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933) 12. Scarpis, V.: sui determinanti di valore massimo. Rend. R. Inst. Lombardo Sci. e Lett. 31(2), 1441–1446 (1898) 13. Seberry, J., Yamada, M.: Hadamard matrices, sequences, and block designs. In: Dinitz, J.H., Stinson, D.R. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 431–560. Wiley, New York (1992) 14. Sylvester, J.J.: Thoughts on inverse orthogonal matrices, simulataneous sign successions, and tesselated pavements in two or more colours, with applications to Newton’s rule, ornamental tilework, and the theory of numbers. Phil. Mag. 34(4), 461–475 (1867) 15. Wallis, J.S.: Combinatorial matrices. Ph.D. Thesis, La Trobe University (1971) 16. Wallis, J.S.: A skew-Hadamard matrix of order 92. Bull. Aust. Math. Soc. 5, 203–204 (1971) 17. Wallis, J.S.: Hadamard matrices. In: Wallis, W.D., Street, A.P., Wallis, J.S. (eds.) Combina- torics: Room Squares, Sum-Free Sets and Hadamard Matrices. Lecture Notes in Mathematics. Springer, Berlin (1972) 18. Wallis, J.S., Whiteman, A.L.: Some classes of Hadamard matrices with constant diagonal. Bull. Aust. Math. Soc. 7, 223–249 (1972) 19. Williamson, J.: Hadamard’s determinant theorem and the sum of four squares. Duke Math. J. 11, 65–81 (1944) Twin Bent Functions and Clifford Algebras

Paul C. Leopardi

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

2m Abstract This paper examines a pair of bent functions on Z2 and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are deﬁned by the properties of a canonical basis for the real representation of the Clifford algebra Rm;m: Some other necessary conditions are also brieﬂy examined.

Keywords Clifford algebras • Bent functions • Hadamard difference sets • Strongly regular graphs

1 Introduction

A recent paper [11] constructs a sequence of edge-coloured graphs m .m > 1/ with two edge colours, and makes the conjecture that for m > 1; there is an automorphism of m that swaps the two edge colours. This conjecture can be reﬁned into the following question.

Question 1.1. Consider the sequence of edge-coloured graphs m .m > 1/ as deﬁned in [11], each with red subgraph mŒ1; and blue subgraph mŒ1: For which m > 1 is there an automorphism of m that swaps the subgraphs mŒ1 and mŒ1? Note that the existence of such an automorphism automatically implies that the subgraphs mŒ1 and mŒ1 are isomorphic. Considering that it is known that mŒ1 is a strongly regular graph, a more general question can be asked concerning such graphs.

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. P.C. Leopardi () Mathematical Sciences Institute, The Australian National University, Canberra, ACT, Australia e-mail: [email protected]

© Springer International Publishing Switzerland 2015 189 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_15 190 P.C. Leopardi

First, we recall the relevant definition. Definition 1.1 ([2, 3, 15]). A simple graph of order v is strongly regular with parameters .v; k;;/if • each vertex has degree k; • each adjacent pair of vertices has common neighbours, and • each nonadjacent pair of vertices has common neighbours. Now, the more general question. Question 1.2. For which parameters .v; k;;/is there an an edge-coloured graph on v vertices, with two edge colours, red (with subgraph Œ1) and blue (with subgraph Œ1), such that the subgraph Œ1 is a strongly regular graph with parameters .v; k;;/;and such that there exists an automorphism of that swaps Œ1 with Œ1? Remark 1.1. Since the existence of such an automorphism implies that Œ1 and Œ1are isomorphic, this implies that Œ1is also a strongly regular graph with the same parameters as Œ1: Questions 1.1 and 1.2 were asked (in a slightly different form) at the workshop on “Algebraic design theory with Hadamard matrices” in Banff in July 2014. Further generalization gives the following questions. Question 1.3. Given a positive integer c >1;for what parameters .v; k;;/does there exist a ck regular graph on v vertices that can be given an edge colouring with c colours, such that the edges corresponding to each color form a .v; k;;/strongly regular graph? For what parameters is the c-edge-coloured ck regular graph unique up to isomorphism? Remark 1.2. This question appears on MathOverflow [9], and is partially answered by Dima Pasechnik and Padraig Ó Catháin, specifically for the case where the ck regular graph is the complete graph on v D ck C 1 vertices. See the relevant papers by van Dam [6], van Dam and Muzychuk [7], and Ó Catháin [13]. These partial answers do not apply to the specific case of Question 1.1 because the graph m is not a complete graph when m >1: Question 1.4. For which parameters .v; k;;/ does the edge-coloured graph from Question 1.3 have an automorphism that permutes the corresponding strongly regular subgraphs? Which finite groups occur as permutation groups in this manner (i.e. as the group of permutations of strongly regular subgraphs of such an edge- coloured graph)?

This paper examines some of the necessary conditions for the graph m to have an automorphism as per Question 1.1. Questions 1.2–1.4 remain open for future investigation. Considering that mŒ1 is a strongly regular graph, the ﬁrst necessary condition is that mŒ1 is also a strongly regular graph, with the same parameters. This is Twin Bent Functions and Clifford Algebras 191 proven as Theorem 5.2 in Sect. 5. Some other necessary conditions are addressed in Sect. 6.

2 A Signed Group and Its Real Monomial Representation

The following deﬁnitions and results appear in the paper on Hadamard matrices and [11], and are presented here for completeness, since they are used below. Further details and proofs can be found in that paper, unless otherwise noted. 1CpCq pCq The signed group Gp;q of order 2 is extension of Z2 by Z2 ,deﬁnedby the signed group presentation

Gp;q WD efkg .k 2 Sp;q/ j

2 1. <0/; 2 1. >0/; efkg D k efkg D k

efjgefkg Defkgefjg .j ¤ k/ ;

where Sp;q WD fq;:::;1;1;:::;pg: The following construction of the real monomial representation P.Gm;m/ of the group Gm;m is used in [11]. The 2 2 orthogonal matrices Ä Ä : :1 ; E1 WD 1: E2 WD 1: generate P.G1;1/; the real monomial representation of group G1;1: The cosets of f˙IgÁZ2 in P.G1;1/ are ordered using a pair of bits, as follows.

0 $ 00 $f˙Ig;

1 $ 01 $f˙E1g;

2 $ 10 $f˙E2g;

3 $ 11 $f˙E1E2g:

For m >1, the real monomial representation P.Gm;m/ of the group Gm;m consists of matrices of the form G1 ˝ Gm1 with G1 in P.G1;1/ and Gm1 in P.Gm1;m1/: The cosets of f˙IgÁZ2 in P.Gm;m/ are ordered by concatenation of pairs of bits, where each pair of bits uses the ordering as per P.G1;1/; and the pairs are ordered as follows.

0 $ 00:::00$f˙Ig;

˝.m1/ 1 $ 00:::01$f˙I.2/ ˝ E1g; 192 P.C. Leopardi

˝.m1/ 2 $ 00:::10$f˙I.2/ ˝ E2g; :::

2m ˝m 2 1 $ 11:::11$f˙.E1E2/ g:

m m (Here I.2/ is used to distinguish this 22 unit matrix from the 2 2 unit matrix I.) In this paper, this ordering is called the Kronecker product ordering of the cosets of f˙Ig in P.Gm;m/: We recall here a number of well-known properties of the representation P.Gm;m/:

Lemma 2.1. The group Gm;m and its real monomial representation P.Gm;m/ satisfy the following properties.

1. Pairs of elements of Gm;m [and therefore P.Gm;m/] either commute or anti- commute: for g; h 2 Gm;m; either hg D gh or hg Dgh: T T 2. The matrices E 2 P.Gm;m/ are orthogonal: EE D E E D I: 3. The matrices E 2 P.Gm;m/ are either symmetric and square to give I or skew and square to give I: either ET D E and E2 D IorET DE and E2 DI: Taking the positive signed element of each of the 22m cosets listed above deﬁnes a transversal of f˙Ig in P.Gm;m/ which is also a monomial basis for the real representation of the Clifford algebra Rm;m in Kronecker product order. In this paper, we call this ordered monomial basis the positive signed basis of P.Rm;m/: For example, .I; E1; E2; E1E2/ is the positive signed basis of P.R1;1/: Note: Any other choice of signs will give a different transversal of f˙Ig in P.Gm;m/; and hence an equivalent ordered monomial basis of P.Rm;m/; but we choose positive signs here for deﬁniteness.

Definition 2.1. We define the function m W Z22m ! P.Gm;m/ to choose the corresponding basis matrix from the positive signed basis of P.Rm;m/; using the Kronecker product ordering. This ordering also defines a corresponding function on 2m Z2 ; which we also call m: For example,

1.0/ D 1.00/ D I;1.1/ D 1.01/ D E1; 1.2/ D 1.10/ D E2;1.3/ D 1.11/ D E1E2:

3 Two Bent Functions

2m We now define two functions, m and m on Z2 ; and show that both of these are bent. First, recall the relevant definition. m Definition 3.1 ([8, p. 74]). A Boolean function f W Z2 ! Z2 is bent if its Hada- mard transform has constant magnitude. Specifically: Twin Bent Functions and Clifford Algebras 193

m 1. The Sylvester Hadamard matrix Hm; of order 2 ; is deﬁned by Ä 11 H1 WD ; 1

Hm WD Hm1 ˝ H1; for m >1:

m 2. For a Boolean function f W Z2 ! Z2; deﬁne the vector f by

m f WD Œ.1/f Œ0;.1/f Œ1;:::;.1/f Œ2 1T ;

wherethevalueoff Œi; i 2 Z2m is given by the value of f on the binary digits of i: m 3. In terms of these two deﬁnitions, the Boolean function f W Z2 ! Z2 is bent if

T jHmf jDCŒ1;:::;1 :

for some constant C:

The first function m is defined and shown to be bent in [11]. We repeat the definition here.

Definition 3.2. We use the basis element selection function m of Definition 2.1 to 2m define the sign-of-square function m W Z2 ! Z2 as ( 1 . /2 . / $ m i DI m i WD 2 0 $ m.i/ D I;

2m for all i in Z2 .

Remark 3.1. Property 3 from Lemma 2.1 ensures that m is well deﬁned. Also, since each m.i/ is orthogonal, m.i/ D 1 if and only if m.i/ is skew. From the property of Kronecker products that .A ˝ B/T D AT ˝ BT ; it can be 2m shown that m can also be calculated from i 2 Z2 as the parity of the number of occurrences of the bit pair 01 in i; i.e. m.i/ D 1 if and only if the number of 01 pairs is odd. Alternatively, for i 2 Z22m ;m.i/ D 1 if and only if the number of 1 digits in the base 4 representation of i is odd. The following lemma is proven in [11]. 2m Lemma 3.1. The function m is a bent function on Z2 .

The basis element selection function m also gives rise to a second function, m on Z22m : 2m Deﬁnition 3.3. We deﬁne the non-diagonal-symmetry function m on Z22m and Z2 as follows. 2 For i in Z2: 194 P.C. Leopardi ( 1 if i D 10; so that 1.i/ D˙E2; 1.i/ WD 0 otherwise:

2m2 For i in Z2 :

m.00 ˇ i/ WD m1.i/;

m.01 ˇ i/ WD m1.i/;

m.10 ˇ i/ WD m1.i/ C 1;

m.11 ˇ i/ WD m1.i/; where ˇ denotes concatenation of bit vectors, and is the sign-of-square function, as above.

It is easy to verify that m.i/ D 1 if and only if m.i/ is symmetric but not diagonal. This can be checked directly for 1: For m >1it results from properties of the Kronecker product of square matrices, specifically that .A ˝ B/T D AT ˝ BT ; and that A ˝ B is diagonal if and only if both A and B are diagonal. The first main result of this paper is the following. 2m Theorem 3.1. The function m is a bent function on Z2 : The proof of Theorem 3.1 uses the following result, due to Tokareva [16], and stemming from the work of Canteaut, Charpin and others [5, Theorem V.4][4, Theorem 2]. The result relies on the following definition. m Definition 3.4. For a bent function f on Z2 the dual function fQ is given by

m=2 fQ.i/ .HmŒf /i DW 2 .1/ :

2m Lemma 3.2 ([16, Theorem 1]). If a binary function f on Z2 can be decomposed 2m2 into four functions f0; f1; f2; f3 on Z2 as

f .00 ˇ i/ DW f0.i/; f .01 ˇ i/ DW f1.i/;

f .10 ˇ i/ DW f2.i/; f .11 ˇ i/ DW f3.i/; where all four functions are bent, with dual functions such that fQ0 CfQ1 CfQ2 CfQ3 D 1; then f is bent.

Proof of Theorem 3.1. In Lemma 3.2,setf0 D f3 WD m1; f1 D m1; f2 D m1 C 1: Clearly, fQ0 D fQ3: Also, fQ2 D fQ1 C 1; since Hm1Œf2 DHm1Œf1: Therefore fQ0 C fQ1 C fQ2 C fQ3 D 1: Thus, these four functions satisfy the premise of Lemma 3.2, as long as both m1 and m1 are bent. It is known that m is bent for all m: Itiseasytoshowthat 1 is bent, directly from its deﬁnition. Therefore m is bent. ut Twin Bent Functions and Clifford Algebras 195

4 Bent Functions and Hadamard Difference Sets

The following well-known properties of Hadamard difference sets and bent functions are noted in [11]. Deﬁnition 4.1 ([8, pp. 10 and 13]). The k-element set D is a .v; k;;n/ difference set in an abelian group G of order v if for every non-zero element g in G; the equation g D di dj has exactly solutions .di; dj/ with di; dj in D: The parameter n WD k: A .v; k;;n/ difference set with v D 4n is called a Hadamard difference set. Lemma 4.1 ([8, Remark 2.2.7] [12, 14]). A Hadamard difference set has parameters of the form

.v; k;;n/ D .4N2;2N2 N; N2 N; N2/ or .4N2;2N2 C N; N2 C N; N2/:

m Lemma 4.2 ([8, Theorem 6.2.2]). The Boolean function f W Z2 ! Z2 is bent if and only if D WD f 1.1/ is a Hadamard difference set. Together, these properties, along with Lemma 3.1 and Theorem 3.1, are used here to prove the following result. 1.1/ 1.1/ Theorem 4.1. The sets m and m are both Hadamard difference sets, with the same parameters

m 2m1 m1 2m2 m1 2m2 .vm; km;m; nm/ D .4 ;2 2 ;2 2 ;2 /:

Proof. Both m and m are bent functions, as per Lemma 3.1 and Theorem 3.1, 1.1/ 1.1/ respectively. Therefore, by Lemma 4.2, both m and m are Hadamard 2m m difference sets. In both cases, the relevant abelian group is Z2 ; with order 4 : Thus in Lemma 4.1 we must set N D 2m1 to obtain that either

m 2m1 m1 2m2 m1 2m2 .vm; km;m; nm/ D .4 ;2 2 ;2 2 ;2 / or m 2m1 m1 2m2 m1 2m2 .vm; km;m; nm/ D .4 ;2 C 2 ;2 C 2 ;2 /:

Since m.i/ D 1 if and only if m.i/ is skew, and m.i/ D 1 if and only if m.i/ is symmetric but not diagonal, not only are these conditions mutually exclusive, but also, for all m > 1; the number of i for which m.i/ D m.i/ D 0 is positive. 2m1 m1 These are the i for which m.i/ is diagonal. Thus km D 2 2 rather than 22m1 C 2m1: The result follows immediately. ut

As a check, the parameters km can also be calculated directly, using the recursive deﬁnitions of each of m and m: 196 P.C. Leopardi

5 Bent Functions and Strongly Regular Graphs

This section examines the relationship between the bent functions m and m and the subgraphs mŒ1 and mŒ1 from Question 1.1. First we revise some known properties of Cayley graphs and strongly regular graphs, as noted in the previous paper on Hadamard matrices and Clifford algebras [11], including the result of Bernasconi and Codenotti [1] on the relationship between bent functions and strongly regular graphs. First we recall a special case of the deﬁnition of a Cayley graph. m Deﬁnition 5.1. The Cayley graph of a binary function f W Z2 ! Z2 is the undirected graph with adjacency matrix F given by Fi;j D f .gi C gj/; for some m ordering .g1; g2;:::/of Z2 : The result of Bernasconi and Codenotti [1] on the relationship between bent functions and strongly regular graphs is the following. m Lemma 5.1 ([1, Lemma 12]). The Cayley graph of a bent function on Z2 is a strongly regular graph with D :

We use this result to examine the graph m: The following two deﬁnitions appear in the previous paper [11] and are repeated here for completeness. 2 m Deﬁnition 5.2. Let m be the graph whose vertices are the n D 4 canonical basis matrices of the real representation of the Clifford algebra Rm;m, with each edge having one of two colours, 1 (red) and 1 (blue):

• Matrices Aj and Ak are connected by a red edge if they have disjoint support and 1 are anti-amicable, i.e. AjAk is skew. • Matrices Aj and Ak are connected by a blue edge if they have disjoint support and 1 are amicable, i.e. AjAk is symmetric. • Otherwise there is no edge between Aj and Ak. We call this graph the restricted amicability/anti-amicability graph of the Clifford algebra Rm;m; the restriction being the requirement that an edge only exists for pairs of matrices with disjoint support. Deﬁnition 5.3. For a graph with edges coloured by 1 (red) and 1 (blue), Œ1 denotes the red subgraph of , the graph containing all of the vertices of ,andall of the red (1) coloured edges. Similarly, Œ1denotes the blue subgraph of . The following theorem is presented in [11].

Theorem 5.1. For all m > 1; the graph mŒ1 are strongly regular, with m 2m1 m1 2m2 m1 parameters vm D 4 ; km D 2 2 ;m D m D 2 2 :

Unfortunately, the proof given there is incomplete, proving only that mŒ1 is 2m1 m1 2m2 strongly regular, without showing why km D 2 2 and m D m D 2 2m1: In this section, we rectify this by proving the following. Twin Bent Functions and Clifford Algebras 197

Theorem 5.2. For all m > 1; both graphs mŒ1 and mŒ1 is strongly regular, m 2m1 m1 2m2 m1 with parameters vm D 4 ; km D 2 2 ;m D m D 2 2 :

Proof. Since each vertex of m is a canonical basis element of the Clifford algebra Rm;m; we can impose the Kronecker product ordering on the vertices, labelling each 1. / Z2m: . ; / . . /; . // vertex A by m A 2 2 The label m a b of each edge m a m b of m depends on a C b in the following way:

m.a; b/ WD m.a C b/ m.a C b/; that is, 8 ˆ <ˆ1; m.a C b/ D 1., m.a C b/ is skew/; . ; / 0; . / . / 0. . / /; m a b D ˆ m a C b D m a C b D , m a C b is diagonal :ˆ 1; m.a C b/ D 1., m.a C b/ is symmetric but not diagonal/:

2m Thus mŒ1 is isomorphic to the Cayley graph of m on Z2 ; and mŒ1 is iso- 2m morphic to the Cayley graph of m on Z2 : Since, by Lemma 3.1 and Theorem 3.1, 2m both m and m are bent functions on Z2 ; Lemma 5.1 implies that both mŒ1 and mŒ1 are strongly regular graphs. It remains to determine the graph parameters. Firstly, vm is the number of vertices, which is 4m: Since mŒ1 is regular, we can determine km by examining one vertex, m.0/: The edges .m.0/; m.b// of mŒ1 are those for which m.b/ D 1; that is, the 1.1/: edges where b is in the Hadamard difference set m Thus, by Theorem 4.1, 2 2m1 m1 m1 km D 2N N D 2 2 ; where N D 2 : Since mŒ1 is a strongly regular graph, it holds that

.vm km 1/m D km.km 1 m/

[15, p. 158] and hence, since m D m; we must have .vm 1/m D km.km 1/: We now note that

2 2 2m2 m1 km.km 1/ D .2N N/.2N N 1/ D .vm 1/.2 2 /;

2m2 m1 so that m D m D 2 2 : Running through these arguments again, with mŒ1 substituted for mŒ1 and m substituted for m; yields the same parameters for mŒ1: ut

Remark 5.1. A more elementary derivation of the value of m for mŒ1 follows. There are km.km 1/ ordered pairs .a; b/ with a ¤ b and m.a/ D m.b/ D 1: 2 2 2 2m2 m1 Since km.km 1/ D .N N/.4N 1/; this gives exactly N N D 2 2 m ordered pairs for each of other 4 1 vertices of mŒ1: 1.1/ Z2m; Also, considering that m is a Hadamard difference set, and for c 2 2 c ¤ 0; consider one of the pairs .a; b/ such that m.a/ D m.b/ D 1 and c D a C b: Thus b D a C c and m.a/ D m.a C c/ D 1: Therefore, the graph mŒ1 contains the edges .m.0/; m.a//; .m.0/; m.b//; .m.c/; m.a//; and .m.c/; m.b//: 198 P.C. Leopardi

Thus, in the graph mŒ1; the vertices m.0/ and m.c/ have the two vertices m.a/ and m.b/ in common. This is true whether or not there is an edge between m.0/ and m.c/: The pair .b; a/ yields the same four edges. Running through all 2 such pairs .a; b/ and using Theorem 4.1 again, we see that m D m D 2N N D 22m2 2m1:

6 Other Necessary Conditions

This section examines two other necessary conditions for the existence of an automorphism of m that swaps mŒ1 with mŒ1: The ﬁrst condition follows.

Theorem 6.1. If an automorphism W m ! m exists that swaps mŒ1 with mŒ1; then there is an automorphism W m ! m that also swaps mŒ1 with mŒ1; leaving m.0/ ﬁxed. Proof. For the purposes of this proof, assume the Kronecker product ordering of the 2m 2m canonical basis elements of Rm;m and deﬁne the one-to-one mapping W Z2 ! Z2 2m such that .m.a// D m. .a// for all a 2 Z2 : The condition that swaps mŒ1 with mŒ1 is equivalent to the condition

m. .a/ C .b// Dm.a C b/; where m is as deﬁned in the proof of Theorem 5.2 above. 2m Let ˚.a/ WD .a/ C .0/ for all a 2 Z2 : Then ˚.a/ C ˚.b/ D .a/ C .b/ for 2m all a; b 2 Z2 ; and therefore

m.˚.a/ C ˚.b// D m. .a/ C .b// Dm.a C b/:

2m Now deﬁne W m ! m such that .m.a// D m.˚.a// for all a 2 Z2 : ut

The second condition is simply to note that if swaps mŒ1 with mŒ1; then for any induced subgraph m and its image ./; the corresponding edges .A; B/ and ..A/; .B// will also have swapped colours. These two conditions were used to design a backtracking search algorithm to find an automorphism that satisfies Question 1.1 or rule out its existence. Two implementations of the search algorithm were coded: one using Python, and a faster implementation using Cython. The source code is available on GitHub [10]. Running the search confirms the existence of an automorphism for m D 1; 2; and 3; but rules it out for m D 4: On an IntelR CoreTM i7 CPU 870 @ 2.93 GHz, the Cython implementation of search for m D 4 takes about 15 h to run. Since this paper was submitted, the author has found a simple proof that an automorphism satisfying Question 1.1 does not exist for m > 4: See arXiv:1504.02827 [math.CO]. Twin Bent Functions and Clifford Algebras 199

Acknowledgements This work was ﬁrst presented at the Workshop on Algebraic Design Theory and Hadamard Matrices (ADTHM) 2014, in honour of the 70th birthday of Hadi Kharaghani. Thanks to Robert Craigen, and William Martin for valuable discussions, and again to Robert Craigen for presenting Questions 1 and 2 at the workshop on “Algebraic design theory with Hadamard matrices” in Banff in July 2014. Thanks also to the Mathematical Sciences Institute at The Australian National University for the author’s Visiting Fellowship during 2014. Finally, thanks to the anonymous reviewer whose comments have helped to improve this paper.

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Máté Matolcsi

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract We describe an approach to the circulant Hadamard conjecture based on Walsh–Fourier analysis. We show that the existence of a circulant Hadamard matrix of order n is equivalent to the existence of a non-trivial solution of a certain homogenous linear system of equations. Based on this system, a possible way of proving the conjecture is proposed.

Keywords Hadamard matrices • Circulant matrices • Walsh–Fourier analysis

1 Introduction

A real Hadamard matrix is a square matrix with ˙1 entries such that the rows (and thus the columns, also) are pairwise orthogonal. A circulant (or cyclic) matrix C is a square matrix which is generated by the cyclic permutations of a row vector, i.e. there exists a vector x D .x1;:::xn/ such that ci;j D xjiC1 for 1 Ä i; j Ä n (the difference being reduced mod n to the set f1;:::;ng; we prefer to use the indices 1;:::;n rather than 0;:::;n 1). It is trivial to check that the 4 4 circulant matrix generated by the row vector .1; 1; 1; 1/ is Hadamard. However, no circulant Hadamard matrix of order larger than 4 is known. The following famous conjecture was made by Ryser [4], more than 50 years ago: Conjecture 1.1 (Circulant Hadamard Conjecture). For n >4there exists no n n circulant real Hadamard matrix.

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. M. Matolcsi () Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, Budapest 1364, Hungary e-mail: [email protected]

© Springer International Publishing Switzerland 2015 201 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_16 202 M. Matolcsi

The first significant result concerning this conjecture was made by Turyn [7] using arguments from algebraic number theory. He proved that if a circulant Hadamard matrix of order n exists then n must be of the form n D 4u2 for some odd integer u which is not a prime-power. The most powerful breakthroughs were later obtained by the “field descent method” of Schmidt [5, 6] and its extensions by Leung and Schmidt [1, 2]. Currently, the smallest open case is n D 4u2 with u D 11;715, and there are less than 1,000 remaining open cases in the range u Ä 1013. In this note we offer a more elementary approach to the circulant Hadamard conjecture, based on Walsh–Fourier analysis.

2 A Walsh–Fourier Approach

The approach described in this note is inspired by the results of [3], where a Fourier analytic approach to the problem of mutually unbiased bases (MUBs) was presented. The basic idea is that the Fourier transform is capable of turning non- linear conditions into linear ones. We briefly introduce the necessary notions and notations here. Let Z2 denote n the cyclic group of order 2,andletG D Z2. An element of G will be regarded as a column vector of length n whose entries are ˙1. And vice versa, each such column vector with ˙1 entries will be regarded as an element of G . Accordingly, an n n matrix A containing ˙1 entries will be regarded as an n-element subset of G ,thecolumnsofA being the elements. We will use (Walsh)–Fourier analysis n on G .LetGO denote the dual group. Then GO is isomorphic to Z2 andanelement of GO will be identified with a row vector containing 0-1 entries. The action of a character D .1;:::n/ 2 GO on an element x D .x1;:::xn/ 2 G is defined 1 n as .x/ D x D x1 :::xn . We will also use the notation GO0 for the subgroup of elements 2 GO such that 1 C 2 CCn Á 0 (mod 2). In this note we will only use a few elements of discrete Fourier analysis on G and GO, asP follows. For any function h W G ! C its Fourier transform is defined as O./ . / GO GO C h D x2G h x x Pfor all 2 . For a function f W ! its Fourier transform is defined as fO.x/ D f ./x for all x 2 G . The convolution of two functions 2GO P ; GO C ./ . / . / f g W ! is defined as f g D 2GO f g . Applying these definitions it is straightforward to verify that fb g.x/ D fO.x/gO.x/ for every x 2 G .Notealso GO Zn ; GO that is isomorphic to 2, thus D C for each P2 , and therefore the ./ . / . / convolution of f and g can also be written as f g D 2GO f C g (we will use this observation in Eq. (8)below). Let A be any n n matrix containing ˙1 entries, and let a1;:::;an denote the columns of A. As explained above, we identify A with the subset fa1;:::;angG , and actually further identify it with the indicator function of this subset. Therefore,P O ./ n the Fourier transform of (the indicator function of) A is given as A D jD1 aj . Noticeherethat A Walsh–Fourier Approach to Circulant Hadamards 203

Xn 2 jAO./j D .aj=ak/ ; (1) j;kD1 where quotient aj=ak is understood coordinate-wise, i.e. a=b D .a1=b1;:::;an=bn/. (As long as we work with ˙1 entries the operation division can be replaced by multiplication, but we prefer to use division in the notation because it can also be used in the more general context of complex Hadamard matrices.) To illustrate the use of the Fourier transform AO ./, let me include here a neat proof of the fact that an n n Hadamard matrix can only exist if 4 divides n.There is an easy combinatorial proof of this fact, but the Fourier proof is also very elegant. Proposition 2.1. If an n n real Hadamard matrix exists, then 4 divides n, or n D 1; 2. Proof. Let H be an n n real Hadamard matrix. If n >1,thenn must clearly be even. Assume 2jn,butn is not divisible by 4. ;::: As described above, the columns h1 hn of H can beP regarded as elements of G Zn 0 1 GO O ./ n D 2 and for any vector 2 we have H D jD1 hj ,and

Xn 2 jHO ./j D .hj=hk/ : (2) j;kD1

Clearly, jHO ./j2 0 for all . However, consider the element D .1;1;:::;1/. On the right-hand side of (2), within the summation we have .hj=hk/ D 1 if j D k, and .hj=hk/ D1 if j ¤ k (here we use the fact that 4 does not divide n). Therefore, the right-hand side evaluates to nn.n1/ Dn.n2/, which is negative if n >2, a contradiction. ut

Let us now turn to circulant Hadamard matrices. Assume u D .u1;:::un/ is a ˙1 vector which generates a circulant Hadamard matrix H. Consider the function

M./ D u (3)

n where ranges over GO D Z2.Letj 2 GO denote the element with an entry 1 at coordinate j, and all other entries being 0. Note that M.j/ D uj. We have the following properties of the function M:

n M./ D˙1 for all 2 Z2; and M.0/ D 1: (4)

This is trivial. n For all d D 1;:::;n=2,andall 2 Z2 we have X M. C j C k/ D 0: (5) jkDd.mod n/ 204 M. Matolcsi P n 0 This is a consequence of the cyclic orthogonality property: jD1 ujujCd D . Spelling it out:

X Xn Xn CjCjCd M. C j C k/ D u D u ujujCd D 0: jkDd.mod n/ jD1 jD1

The aim is to get a contradiction from the facts (4), (5)forn >4.Ifwejust consider the conditions (5), and regard each M./ as a real variable,thenwehavea n n n homogenous system of linear equations with 2 variables and 2 2 linear constraints. We will prove that this is an equivalent formulation of the circulant Hadamard conjecture, i.e. the existence of any non-trivial solution to this linear system of equations implies the existence of a circulant Hadamard matrix of order n. We will ﬁrst need some intermediate lemmas. Lemma 2.1. The circulant Hadamard conjecture is true for n if and only if the n-variable equation

0 12 Xn1 Xn @ A ujujCd D 0 (6) dD1 jD1 admits no such solution where each variable uj assumes ˙1 value. Proof. This is trivial. ut While the above lemma is trivial, it can be combined with the system of equations (5). Let S W GO0 ! R denote the function deﬁned by the coefﬁcients on the left-hand side of (6), i.e.

0 12 Xn1 Xn X @ A ujujCd D S. /u : (7) dD1 jD1

2 1 Note that we have used the simpliﬁcation uj D (for each j) on the left-hand side, so that indeed only monomials of the form u withP 2 GO will appear on the right. Note that the right-hand side can also be written as S. /M. /. Similar to (5) we can now write a system of linear equations involving S:if u generates a cyclic Hadamard matrix, then M./ D u satisﬁes the following equations: X X n M. C /S. / D M./ M. /S. / D 0 for all 2 Z2: (8)

Keep in mind here that we will regard the values of M as real variables (disregarding the fact that M must be ˙1-valued). Therefore, there are 2n variables A Walsh–Fourier Approach to Circulant Hadamards 205 and we have also 2n the linear equations [one for each 2 GO,asgivenin(8)]. This linear system leads to a coefficient matrix of size 2n 2n.Anyrowinthe coefficient-matrix will contain the same numbers S. /, but the position of S. / is n shifted according to the geometry of GO D Z2. We will now show that the existence of a circulant Hadamard matrix of order n is equivalent to the coefficient-matrix being singular. Lemma 2.2. Regard each M./ as a real variable, and consider the homogenous system of linear equations determined by (8). There exists a ˙1 vector u generating a cyclic Hadamard matrix if and only if (8) admits a non-trivial solution M./. Proof. If u generates a cyclic Hadamard matrix, then M./ D u satisfies (8), yielding a non-trivial solution. In the converse direction, assume M./ is a non- trivial solution to (8). That is, M is not identically 0, but we do not assume that M is ˙1-valued. Notice that the left-hand side of (8) is the convolution M S of the functions M and S on the group GO. This means that the convolution M S Á 0 on GO. Taking Fourier transform we conclude that M1 S.x/ D MO .x/SO.x/ D 0 for every x 2 G .AsM is not identically zero, its Fourier transform cannot be identically zeroP either. Hence there exists an u 2 G such that MO .u/ ¤ 0 and therefore SO.u/ D

S. /u D 0.By(7) this means exactly that there exist a solution u to the Eq. (6). ut We can now prove that the linear system of equations (5) is also an equivalent formulation of the circulant Hadamard conjecture. Lemma 2.3. Regard each M./ as a real variable, and consider the homogenous system of linear equations determined by (5). The circulant Hadamard conjecture is true for n if and only if this system of equations has full rank, i.e. the only solution is M./ D 0 for each . Proof. One direction is trivial: if u generates a circulant Hadamard matrix, then M./ D u is a non-trivial solution to (5). Conversely, if there exists a non-trivial solution M./ of (5), then M is a fortiori a solution of (8), because each equation in (8) is a linear combination of some equations in (5). Therefore a circulant Hadamard matrix exists by Lemma 2.2. ut While all the results above are fairly trivial, they do have some philosophical advantages. First, we can rest assured that Ryser’s circulant Hadamard conjecture can be proved or disproved in this manner—we have not lost any information by setting up the system (5). Second, the circulant Hadamard conjecture is a nonexistence conjecture, which can now be transformed to an existence result (i.e. it is enough to exhibit a witness which proves the non-existence of circulant Hadamard matrices): Corollary 2.1. The circulant Hadamard conjecture is true for n if and only if there existrealweightsc;d such that 206 M. Matolcsi 0 1 X X @ A c;d M. C j C k/ D M.0/ (9) ;d jkDd.mod n/

Proof. If such weights exist, then (5) cannot admit a solution in which M.0/ D 1, and hence there cannot exist a circulant Hadamard matrix of order n.Conversely,if such weights do not exist, then the linear system (5) does not have full rank, so a circulant Hadamard matrix of order n exists by Lemma 2.3. ut Therefore we are left with the “simple” task of exhibiting a witness (a set of weights c;d) for each n. It is possible to obtain such witnesses by computer for small values of n,i.e.n D 8; 12; 16; 20; 24. The problem is that there are always an infinite number of witnesses (a whole affine subspace of them with large dimension), and one should somehow select the “nicest” one, which could be generalized for any n. It is natural to exploit the invariance properties of the problem as follows. If M./ is a non-trivial solution to (5), then so is M ./ D M..// where is any cyclic permutation of the coordinates. We can therefore define equivalence classes in GO, regarding 1 and 2 equivalent if they are cyclic permutations of each other. After averaging we can then assume that the required weights c;d are constant on equivalence classes. Furthermore, if 1 Ä k Ä n 1 is relatively prime to n, then multiplication by k defines an automorphism of the cyclic group Zn.We can regard 1 and 2 equivalent if a coordinate transformation corresponding to multiplication by some k transforms one to the other. Similarly, we can regard d1 and d2 equivalent if GCD(d1; n)=GCD(d2; n). After averaging again, we can assume that the required witness weights c;d depend only on the equivalence class of and that of d. However, such restrictions still do not determine the weights c;d uniquely, and still the witnesses form an affine subspace of large dimension. It is also easy to see that we mayP restrict our attention without loss of generality n to the subgroup GO0 Df 2 GO W Á 0.mod 2/g, because all the terms on the jD1 j P GO GO n left-hand side of (5)stayin 0 if 2 0. We will call jD1 j the weight of ,and denote it by jj. In the last section of this note we will consider symmetric polynomials of the variables uj, i.e. expressions of the form

Xn X dwM./: (10) 2jwD0 jjDw

That is, only 2 GO0 are considered in the sum, and the coefﬁcient of M./ depends on the weight of only. It is trivial to see that if M./ D u then (10) is a symmetric polynomial of u1;:::;un. Expressions of the form (10) constitute a vector space of n dimension 2 C 1, a natural basis of which is given by the single-weight expressions X M./; w D 0;2;4;:::n: (11) jjDw A Walsh–Fourier Approach to Circulant Hadamards 207

One way to generate an expression of the form (10) using the Eq. (5)isthe following:

X Xn=2 X M. C j C k/; w D 0;2;4;:::n: (12) jjDw dD1 jkDd.mod n/

Lemma 2.4. If 4 divides n, then the dimension of the subspace spanned by the n expressions (12) in the vector space of the expressions of the form (10) is 2 C 1 if 2 n 2 n ¤ 4u , while it is 2 if n D 4u . Proof. For any 2 Ä w Ä n 2 the left-hand side of the expression (12) will contain variables M./ where the weight jj is w2;w or wC2.Forw D 0 we will have ’s with weight 0; 2, while for w D n we will have ’s with weight n 2;n. Therefore, n it is easy to express (12) in the basis (11) explicitly, as a vector of length 2 C 1 with only three non-zero coordinates for 2 Ä w Ä n 2 and only two non-zero coordinates for w D 0 and w D n. This leads to a tri-diagonal coefﬁcient matrix n 2 n 2 whose rank is 2 C 1 if n ¤ 4u , while it is 2 if n D 4u . The explicit calculations are left to the reader. ut This lemma leads to the following well-known corollary: Lemma 2.5. If there exists a cyclic Hadamard matrix of order n, then n must be an even square number, n D 4u2. Proof. By Proposition 2.1 n must be divisible by 4. If n ¤ 4u2, then by Lemma 2.4 we see that the expressions (12) generate the whole space of symmetric polynomials given by (10). In particular, the single variable M.0/ (being a symmetric polynomial in itself) is also in this subspace, so we conclude that there exists an expansion of the form

X Xn=2 X cw M. C j C k/ D M.0/; (13) jjDw dD1 jkDd.mod n/ which is a special case of (9). ut One might object that this is a very difficult way of proving a very easy statement. However, it does have some advantages. First, it rhymes very well with (9)and the strategy described in the paragraphs after Lemma 2.1. Namely, put the ’s and the d’s into some equivalence classes and look for a solution to (9) such that the coefficients depend only on the equivalence classes. Second, it “nearly” works n even if n is a square: the span of the expressions (12) has dimension 2 . One could therefore hope for the following strategy to work. Let us call a linear combination on the left-hand side of (13) “trivial”. If we could find a non-trivial linear combination (9) such that the result is of the form (10), then it is “very likely” that the dimension 208 M. Matolcsi

n of the span would increase to 2 C 1, which would complete the proof of the general case. It is not at all clear whether such “magic” non-trivial linear combination is easy to ﬁnd for general n, but it is not out of the question.

Acknowledgements The author was supported by OTKA grant No. 109789 and by ERC-AdG 321104.

References

1. Leung, K.H., Schmidt, B.: The field descent method. Des. Codes Crypt. 36, 171–188 (2005) 2. Leung, K.H., Schmidt, B.: New restrictions on possible orders of circulant Hadamard matrices. Des. Codes Crypt. 64, 143–151 (2012) 3. Matolcsi, M., Ruzsa, I.Z., Weiner, M.: Systems of mutually unbiased Hadamard matrices containing real and complex matrices. Austral. J. Combin. 55, 35–47 (2013) 4. Ryser, H.J.: Combinatorial Mathematics. Wiley, New York (1963) 5. Schmidt, B: Cyclotomic integers and finite geometries. J. Am. Math. Soc. 12, 929–952 (1999) 6. Schmidt, B: Towards Ryser’s conjecture. In: Casacuberta, C., et al. (eds.) Proceedings of the Third European Congress of Mathematics. Progress in Mathematics, vol. 201, pp. 533–541. Birkhuser, Boston (2001) 7. Turyn, R.J.: Character sums and difference sets. Pacific J. Math. 15, 319–346 (1965) A Note on Order and Eigenvalue Multiplicity of Strongly Regular Graphs

A. Mohammadian and B. Tayfeh-Rezaie

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract In this note, we consider a well-known upper bound for the order of a strongly regular graph in terms of the multiplicity of a non-principal eigenvalue of its adjacency matrix.

Keywords Adjacency matrix • Eigenvalue multiplicity • Strongly regular graph

Mathematics Subject Classiﬁcation (2010): 05C50, 05E30

1 Order and Eigenvalue Multiplicity of Strongly Regular Graphs

A strongly regular graph with parameters .n; k;;/, denoted srg.n; k;;/,is a regular graph of order n and valency k such that (i) it is not complete or edgeless, (ii) every two adjacent vertices have common neighbors, and (iii) every two non-adjacent vertices have common neighbors. Strongly regular graphs form an important class of graphs which lie somewhere between highly structured graphs and apparently random graphs. We refer the reader to see Chapter 9 of [1] for a survey and [3] for recent results on parameters of strongly regular graphs. Obviously, complete multipartite graphs with equal part sizes and their complements are trivial examples of strongly regular graphs. In this note, to exclude

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. A. Mohammadian • B. Tayfeh-Rezaie () School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran e-mail: [email protected]; [email protected]

© Springer International Publishing Switzerland 2015 209 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_17 210 A. Mohammadian and B. Tayfeh-Rezaie these examples, we assume that a strongly regular graph and its complement are connected; in other words, we assume, equivalently, that 0<

A A2 . /A . / ; GJn D kJn and G C G C k In D Jn where In and Jn are the nn identity matrix and the nn all one matrix, respectively. It is not hard to see that the eigenvalues of an srg.n; k;;/are 8 ˆ k; with the multiplicity 1I ˆ p <ˆ C n 1 2k C .n 1/. / r D ; with the multiplicity f D p I 2 2 2 ˆ p ˆ n 1 2k C .n 1/. / : s D ; with the multiplicity g D C p ; 2 2 2 where D . /2 C 4.k /. It is well known that the second largest eigenvalue of a graph G is non-positive if and only if the non-isolated vertices of G form a complete multipartite graph. Also, it is a known fact that the smallest eigenvalue of agraphG is at least 1 if and only if G is a vertex disjoint union of some complete graphs. Therefore, for any srg.n; k;;/,wehaver >0and s < 1. The important conditions satisﬁed by the parameters of a strongly regular graph are the Krein condition [5] and the absolute bound [6]: .r C 1/.k C r C 2rs/ 6 .k C r/.s C 1/2;.1/ The Krein condition: .s C 1/.k C s C 2sr/ 6 .k C s/.r C 1/2I .2/ n 6 f .f C 3/=2; .3/ The absolute bound: n 6 g.g C 3/=2: .4/

It was shown in [4] that (3) can be improved to n 6 f .f C 1/=2, unless equality occurs in (1). A similar statement holds for (2) and (4). It is easy to see that equality occurs in (1) for a strongly regular graph if and only if the graph is the pentagon or an srg.n; k;;/with integral eigenvalues in which

8 2 2 ˆ 2.s r/ .s C 2s C 2sr C r/ ˆ n D ; ˆ . 2 /. 2 2 / ˆ s C r s C s r ˆ r.s2 C 2sr C r/ < k D ; 2 2 s C s2 r ˆ s.r C 1/.s C 2s C r/ ˆ D ; ˆ s2 C 2s r ˆ 2 :ˆ r.s C 1/.s C r/ D : s2 C 2s r A Note on Order and Eigenvalue Multiplicity of Strongly Regular Graphs 211

A strongly regular graph with these parameters or the complement of one is called a Smith graph. Since r >0and s < 1, the non-negativity of shows that r > s2C2s. Moreover, we have

.s2 2sr r/.s2 C 2s C 2sr C r/ 2r.r C 1/.s2 C 2sr C r/ f D and g D : .s2 C r/.s2 C 2s r/ .s2 C r/.s2 C 2s r/

By an easy calculation, we find that .s2 C 2sr C r/ .s C r C 1/2 C r2 C r 1 g f D >0: .s2 C r/.s2 C 2s r/

In this note, we improve the aforementioned result of [4]. Lemma 1.1. If, for strongly regular graphs, equality occurs in (1), then either n 6 f .f C 1/=2 or n D f .f C 3/=2, unless the graph is the Clebsch graph, that is, the unique srg.16; 10; 6; 6/. Proof. Let equality occur in (1) for a strongly regular graph. Using (3), we may suppose that f .f C 1/=2 < n < f .f C 3/=2.Since

f .f C 3/ 2r.r C 1/.r s/.s2 C 2s C 2sr C r/.2s3 C 3s2 C r/ n D ; 2 .s2 C r/2.s2 C 2s r/2 we have r < 2s3 3s2.Also,

f .f C 1/ 2r.r C 1/.s2 2sr r/.s3 C 2s2 C 2s3r C 3s2r sr C r2/ n 1 D ; 2 .s2 C r/2.s2 C 2s r/2 and hence

s3 C 2s2 C 2s3r C 3s2r sr C r2 D s2.s C 2/ C r.2s3 C 3s2 s C r/ > 0:

This implies that r > 2s3 3s2 C s. Letting ` D 2s3 C 3s2 C r,wehave

`.s C 1/2 ` D 2s4 C 3s3 s` ` C : 2s.s C 1/2 `

Since s 6 ` 6 1 and is integral, it is straightforward to see that s D2.From 2s3 3s2 C s 6 r 6 2s3 3s2 1 and the integrality of n,wefindthatr D 2 and so the graph is srg.16; 10; 6; 6/. ut Note that equality occurs in (1) for a strongly regular graph if and only if equality occurs in (2) for its complement. So, by Lemma 1.1, we obtain the following result. 212 A. Mohammadian and B. Tayfeh-Rezaie

Theorem 1.1. For any strongly regular graph, one of the following holds. f .f C 1/ g.g C 1/ (i) n 6 min ; . 2 2 f .f C 3/ g.g C 3/ (ii) n ; . D min 2 2 (iii) The graph or its complement is the Clebsch graph. Remark 1.1. Let us consider the equality cases in Theorem 1.1. There are only three known examples of strongly regular graphs satisfying n D f .f C 3/=2; these are the pentagon, the Schläfli graph, and the McLaughlin graph. There are inﬁnitely many feasible parameters of strongly regular graphs with n D f .f C 1/=2. It is not hard to check that a strongly regular graph with n D f .f C 1/=2 and s D2 has parameters k D 2f 2, D f 1,and D 4.Byaresultof[2], any such strongly regular graph with f ¤ 7 is a triangular graph, that is, the line graph of the complete graph of order f C 1. The problem of characterizing strongly regular graphs with n D f .f C 1/=2 which is posed in [4] remains unsolved.

Acknowledgements The research of the ﬁrst author was in part supported by a grant from IPM (No. 91050405).

References

1. Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Universitext. Springer, New York (2012) 2. Chang, L.C.: The uniqueness and nonuniqueness of the triangular association schemes. Sci. Record. 3, 604–613 (1959) 3. Elzinga, R.J.: Strongly regular graphs: values of and for which there are only ﬁnitely many feasible .v; k;;/. Electron. J. Linear Algebra 10, 232–239 (2003) 4. Neumaier, A.: New inequalities for the parameters of an association scheme. In: Combinatorics and Graph Theory. Lecture Notes in Mathematics, vol. 885, pp. 365–367. Springer, Berlin (1981) 5. Scott, Jr., L.L.: A condition on Higman’s parameters. Notices Am. Math. Soc. 20 (A-97), Abstract 701-20-45 (1973) 6. Seidel, J.J.: Strongly regular graphs. In: Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 38, pp. 157–180. Cambridge University Press, Cambridge (1979) Trades in Complex Hadamard Matrices

Padraig Ó Catháin and Ian M. Wanless

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c ¤ 1 to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order n and show that they all contain at least n entries. We conjecture that all trades in complex Hadamard matrices contain at least n entries.

Keywords Hadamard matrix • Trade • Rank

Mathematics Subject Classiﬁcation (2010): 05B20, 15B34

1 Introduction

A complex Hadamard matrix of order n is an n n complex matrix with unimodular entries which satisﬁes the matrix equation

HH D nIn;

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. P. Ó Catháin ()•I.M.Wanless School of Mathematical Sciences, Monash University, Melbourne, VIC 3800, Australia e-mail: [email protected]; [email protected]

© Springer International Publishing Switzerland 2015 213 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_18 214 P. Ó Catháin and I.M. Wanless

where H is the conjugate transpose of H and In is the n n identity matrix. If the entries are real (hence ˙1), the matrix is Hadamard. The notion of a trade is well known in the study of t-designs and Latin squares [1]. For a complex Hadamard matrix we deﬁne a trade to be a set of entries which can be altered to obtain a different complex Hadamard matrix of the same order. In other words, a set T of entries in a complex Hadamard matrix H is a trade if there exists another complex Hadamard matrix H0 such that H and H0 disagree on every entry in T but agree otherwise. If H is a real Hadamard matrix, we insist that H0 is also real. Example 1.1. The eight shaded entries in the Paley Hadamard matrix below form a trade. 0 1 C CCCCCCC B C B CCCCC B C B CC CCC B C B C B CCC CC B C B CCCCC B C B C CCCC @ CCCCA CCCC

If each of the shaded entries is replaced by its negative, the result is another Hadamard matrix. We use the word switch to describe the process of replacing a trade by a new set of entries (which must themselves form a trade). In keeping with the precedent from design theory, our trades are simply a set of entries that can be switched. Information about what they can be switched to does not form part of the trade (although it may be helpful in order to see that something is a trade). For real Hadamard matrices there can only be one way to switch a given trade, since only two symbols are allowed in the matrices and switching must change every entry in a trade. However, for complex Hadamard matrices there can be more than one way to switch a given trade, as our next example shows. Example 1.2. Let u be a nontrivial third root of unity. The following matrix is a 7 7 complex Hadamard matrix. The shaded entries again form a trade; they can be multiplied by an arbitrary complex number c of modulus 1 to obtain another complex Hadamard matrix. This matrix is due originally to Petrescu [9], and is available in the online database [2]. 0 1 1111111 B C B 1 u u u2 1 1 u C B C B 1 u u 1 u2 1 u C B C B 1 2 1 1 C B u u u u C B 1 1 2 1 C B u u u u C @ 1 1 1 u uuu2 A 1 u u 1 1 u2 u Trades in Complex Hadamard Matrices 215

The size of a trade is the number of entries in it. We say that a trade is rectangular if the entries in the trade form a submatrix that can be switched by multiplying all entries in the trade by some complex number c ¤ 1 of unit modulus. It will follow from Lemma 2.1 that the value of c is immaterial; if one value works, then they will all work. In a complex Hadamard matrix each row and column is a rectangular trade. Thus there are always 1n and n1 rectangular trades. Similarly, we may exchange any pair of rows to obtain another complex Hadamard matrix. In the real case, the rows that we exchange necessarily differ in exactly half the columns, so this reveals n n a 2 2 rectangular trade (and similarly there are always 2 2 rectangular trades in real Hadamard matrices). Less trivial trades were used by Orrick [8] to generate many inequivalent Hadamard matrices of orders 32 and 36. The smaller of Orrick’s n two types of trades was a 4 4 rectangular trade that he called a “closed quadruple”. Closed quadruples are often but not always present in Hadamard matrices. The trades just discussed all have size equal to the order n of the host matrix. The trade in Example 1.1 is a non-rectangular example with the same property. Trades in real Hadamard matrices and related codes and designs have been studied occasionally in the literature, either to produce invariants to aid with classiﬁcation or to produce many inequivalent Hadamard matrices. See [8]andthe references cited there. In the complex case, trades are related to parameterising complex Hadamard matrices, some computational and theoretical results are surveyed in [10]. Throughout this note we will assume that H D Œhij is a complex Hadamard matrix of order n. We will use ri and cj to denote the i-th row and j-th column of H, respectively. If B is a set of columns, then ri;B denotes the row vector which is equal to ri on the coordinates B and zero elsewhere. We use B for the complement of the set B.

2 Hadamard Trades ˝ ˛ We start with a basic property of trades. We use ; for the standard Hermitian inner product under which rows of a complex Hadamard matrix are orthogonal. Lemma 2.1. Let T be a subset of the entries of a complex Hadamard matrix H. Let c ¤ 1 be a complex number of unit modulus. 1. Suppose that T can be switched by multiplying its entries by c. Let B be the set of columns in which row ri of H contains elements of T. If rj is a row of H that contains no elements of T, then ri;B is orthogonal to rj;B. 2. Suppose that T forms a rectangular submatrix of H with rows A and columns B. Then T can be switched by multiplying its entries by c if and only if ri;B is orthogonal to rj;B for every ri 2 A and rj … A. 216 P. Ó Catháin and I.M. Wanless

Proof. First, since the rows of H are orthogonal, we have that ˝ ˛ ˝ ˛ ˝ ˛ 0 ; ; ; : D ri rj D ri;B rj;B C ri;B rj;B

Now, multiplying the entries in T by c, we see that ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ 0 ; ; ; ; : D cri;B rj;B C ri;B rj;B D c ri;B rj;B C ri;B rj;B ˝ ˛ Subtracting, we find that .c 1/ ri;B; rj;B D 0. Given that c ¤ 1 the first claim of the Lemma follows. We have just shown the necessity of the condition in the second claim. To check sufficiency we note that the above argument is reversible and shows that ri 2 A and rj … A will be orthogonal after multiplication of the entries of T by c.Sowejust have to verify that any two rows ri; rk in A will be orthogonal. This follows from ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ ; ; ; ; ; ; 0: cri;B crk;B C ri;B rk;B Djcj ri;B rk;B C ri;B rk;B D ri;B rk;B C ri;B rk;B D

ut Note that the value of c plays no role in Lemma 2.1. Also, Part 1 of the lemma implies that in a real Hadamard matrix any trade which does not intersect every row must use an even number of entries from each row. The same is not true for trades in complex Hadamard matrices (see [2] for examples). It is of interest to consider the size of a smallest possible trade. For (real) Hadamard matrices of order n we show that arbitrary trades have size at least n. Equality is achievable in a variety of ways, as discussed above. However, we find a restriction that must be obeyed by any trade achieving equality. Then we show that in the general case rectangular trades have size at least n. The question for arbitrary trades in complex Hadamard matrices remains open. Theorem 2.1. Let H be a (real) Hadamard matrix of order n. Any trade in H has size at least n. If T is any trade of size n in H, then there are divisors d and e of n such that T contains either 0 or d entries in each row of H and either 0 or e entries in each column of H. Moreover, d is even or d D 1. Likewise, e is even or e D 1. Proof. Suppose that H differs from a Hadamard matrix H0 in a trade T of at most n entries. Without loss of generality, we assume that H is normalised, that the first row of H contains d differences between H and H0, and that these differences occur in the first d columns. We also assume that all differences between H and H0 occur in the first r rows, with each of those rows having at least d differences in them. The case r D n is trivial, so we assume that r < n in the remainder of the proof. By assumption there are at least rd entries in T,sord Ä n. Now consider the submatrix S of H formed by the first d columns and the last n r rows. By Lemma 2.1,we know that each row of S is orthogonal to the all ones vector. It follows that d is even and S contains .n r/d=2 negative entries. The first column of S consists entirely of ones so, by the pigeon-hole principle, some other column of S must contain at least Trades in Complex Hadamard Matrices 217

.n r/d nd n n Ä D (1) 2.d 1/ 2.d 1/ 2 negative entries. This column of H is orthogonal to the ﬁrst column, so we must have equality in (1). It follows that n D rd and each of the ﬁrst r rows contains exactly d entries in T. Columns have similar properties, by symmetry. ut Corollary 2.1. In a (real) Hadamard matrix of order n the symmetric difference of any two trades must have size at least n.

Proof. Suppose that H; H1; H2 are distinct (real) Hadamard matrices of order n.Let T1 and T2 be the set of entries of H which disagree with the corresponding entries of H1 and H2, respectively. The symmetric difference of T1 and T2 has cardinality equal to the number of entries of H1 that are different to the corresponding entry of H2. This cardinality is at least n, by Theorem 2.1. ut Example 1.1 is the symmetric difference of two rectangular trades, one 2 4 and the other 4 2. It shows that equality can be achieved in the Corollary. The example also demonstrates that trades of minimal size need not be rectangular. In the notation of Theorem 2.1 it has d D e D 2 and n D 8. Another example is obtained as follows. Let H be any Hadamard matrix and H0 the matrix obtained by swapping two rows of H, then negating one of the rows that was swapped. Let T be the trade consisting of the entries of H which differ from the corresponding entry in H0. It is easy to show that T has d D n=2, e D 1 in the notation of Theorem 2.1. It is also possible to have d D e D 1. If this is the case, then by permuting and/or negating rows we obtain a Hadamard matrix H for which H 2I is also Hadamard, where I is the identity matrix. However this means that

HH> D .H 2I/.H 2I/> D HH> 2H 2H> C 4I:

Hence H C H> D 2I,soH is a skew-Hadamard matrix. Conversely, the main diagonal of any skew-Hadamard matrix is a trade with d D e D 1. Now we consider complex Hadamard matrices. The following lemma is the key step in our proof. The corresponding result for real Hadamard matrices has been obtained by Alon (cf. [5], Lemma 14.6). Alon’s proof can be trivially adapted to deal with complex Hadamard matrices. We include our own independent proof here since we want to extract a characterisation of cases where the bound is tight. Lemma 2.2. Let H be a complex Hadamard matrix of order n, and B a set of b columns of H. If ˛ is a non-zero linear combination of the elements of B, then ˛ has n at least d b e non-zero entries. Proof. Without loss of generality, we can write H in the form Â Ã TU H D VW 218 P. Ó Catháin and I.M. Wanless where T contains the columns in B and the rows in which ˛ is non-zero. We will identify a linear dependence among the rows of U, then use this and an expression for the inner product of r1 and r2 to derive the required result. We assume that there are t non-zero entries ˛i in ˛ and that if t 2 then they obey j˛2jj˛1jj˛ij for 3 n Ä i Ä t. We need to show that t db e. ˝ ˛ For any column cj not in B,wehavethat cj;˛ D 0 since the columns of H are orthogonal. Thus every column of U is orthogonal toP˛, and so there exists a linear t ˛ ˛1 dependence among the rows of U, explicitly: h1j D iD2 i 1 hij,foranyj … B. In particular, this shows that indeed t 2. Since H is Hadamard, we know that all of the hij have absolute value 1,andthat rows of H are necessarily orthogonal: ˝ ˛ ˝ ˛ ˝ ˛ ; ; ; r1 r2 D r1;B r2;B C r1;B r2;B ˝ ˛ ˝ Xt ˛ ; ˛ ˛1 ; D r1;B r2;B C i 1 ri;B r2;B iD2 ˝ ˛ Xt ˝ ˛ ; ˛ ˛1 ; : D r1;B r2;B C i 1 ri;B r2;B iD2 ˝ ˛ ˝ ˛ ˝ ˛ ; 0 ; ; Since r1 r2 D and ri;B r2;B Dri;B r2;B , this means that

˝ ˛ ˝ ˛ Xt ˝ ˛ ˛ ˛1 ; ; ˛ ˛1 ; : 2 1 r2;B r2;B D r1;B r2;B C i 1 ri;B r2;B (2) iD3 ˝ ˛ Now, each inner product ri;B; r2;B is a sum of b complex numbers of modulus one, 1 and j˛i˛1 jÄ1 for i 3. So the absolute value of the right-hand side of (2) is at most .t 1/b. In contrast, the absolute value of the left-hand side of (2)is 1 n ˛2˛ . / . 1/ j 1 j n b n b. It follows that n b Ä t b, and hence t db e. ut Let H be a Fourier Hadamard matrix of order n, and suppose that t j n.Then there exist t rows of H containing only tth roots of unity. Their sum vanishes on all n but t coordinates, so Lemma 2.2 is best possible. On the other hand, if H is Fourier of prime order p, the only vanishing sum of pth roots is the complete one. So in this case, a linear combination of at most t rows will contain at most t zero entries. Theorem 2.2. If H is a complex Hadamard matrix of order n containing an a b rectangular trade T, then ab n. If ab D n, then T is a rank one submatrix of H.

Proof. Without loss of generality, T lies in the first a rows of H.LetPB be the set of theP columns that contain the entries of T. By hypothesis, 1 D 1ÄiÄa ri and . / c D 1ÄiÄa cri;B C ri;B are both orthogonal to the space U spanned by the last n a rows of H. Now consider 1 c, which is zero in any column outside B, but which is not zero since the rows of H are linearly independent. Observe that the Trades in Complex Hadamard Matrices 219 orthogonal complement of U is a-dimensional, and that the initial a rows of H span this space: thus 1 c is in the span of these rows, Lemma 2.2 applies, and ab n. If ab D n, then equality holds in calculations at the end of the proof of Lemma 2.2. In particular, jhri;B; r2;Bij D b for each i, which implies that ri;B is collinear to r2;B. Hence T is a rank one submatrix of H. ut We now give a complete characterisation of the minimal rectangular trades in any complex Hadamard matrix. Theorem 2.3. Let H be a complex Hadamard matrix of order n and T an a b submatrix of H with ab D n. Then T is a rectangular trade if and only if T is rank 1. Proof. Theorem 2.2 shows that any rectangular trade of size n is necessarily rank one. So we need only prove the converse. Without loss of generality, we assume that T is contained in the first a rows and first b columns of H and that H is normalised. Note that this implies that T is anP all ones submatrix. . ;:::; / b Consider D 1 n D iD1 ci, the sum of the first b columns of H.Itis clear that j D b for j 2f1;:::;ag. If we show that j D 0 for a < j Ä n,then Lemma 2.1 will show that T is a trade. We calculate the `2 norm of in two ways: first, via an expansion into orthogonal vectors:

Xb Xb Xb 2 kk2 Dh ci; ciiD hci; ciiDbn: iD1 iD1 iD1 P 2 n 2 1 On the other hand, k k2 D iD1 j ij .Wehavethat i D b for Ä i Ä a.But 2 ab D nb,sojijD0 for all i > a. Applying Lemma 2.1, we are done. ut Corollary 2.2. If T is an a b rank one submatrix of H, then T is a trade if and only if ab D n. Proof. We have that ab n by Theorem 2.2. In the other direction, Lindsay’s Lemma states that the size of a rank one submatrix of a Hadamard matrix of order n is bounded above by n (see Lemma 14.5 of [5]). ut Ryser’s embedding problem is to establish the minimal order, R.a; b/,ofa Hadamard matrix containing an a b submatrix consisting entirely of ones. Any rank one submatrix can be transformed into a submatrix consisting entirely of ones by a sequence of Hadamard equivalence operations. Hence there is a Hadamard matrix of order ab containing an a b rectangular trade if and only if R.a; b/ D ab. Newman [7] showed that R.a; b/ D ab whenever both a; b are orders for which Hadamard matrices exist. Michael [6] showed that R.a; b/ .aC1/b for odd a >1. Thus there are no a b rectangular trades in this case, a conclusion that could also be reached from Theorem 2.1. Michael also showed that if 2a and b=2 are orders of Hadamard matrices then there exists an a b rectangular trade in a Hadamard matrix of order ab. For example, there is a Hadamard matrix of order 48 containing a 6 8 rectangular trade. 220 P. Ó Catháin and I.M. Wanless

3 Open Questions

A Bush type Hadamard matrix of order m2 contains an m m rank one submatrix. Hence there is a Hadamard matrix of order 36 containing a 6 6 rectangular trade. Thus all cases of our ﬁrst question smaller than a D 6, b D 10 are resolved. Question 1. Are there even integers a; b for which there does not exist a Hadamard matrix of order ab containing an a b rectangular trade? On the basis of Theorems 2.1 and 2.2 we are inclined to think that the answer to the following question is negative: Question 2. Can there exist trades of size less than n in an nn complex Hadamard matrix? It would also be nice to know how “universal” the rectangular trades we have studied are. Example 1.1 showed that combinations of rectangular trades can create more complicated trades. By iterating such steps can we build all trades? In other words:

Question 3. Is every trade in a (real) Hadamard matrix a Z2-linear combination of rectangular trades? If so, how does this generalise to the complex case? This work was motivated in part by problems in the construction of compressed sensing matrices [3]. Optimal complex Hadamard matrices for this application have the property that linear combinations of t rows vanish in at most t components. Question 4. Other than Fourier matrices, are their families of Hadamard matrices with the property that no linear combination of t rows contains more than t zeros? Or, if such matrices are rare, describe families in which no linear combination of t rows contain more than f .t/ zeros for some slowly growing function f . We are indebted to Prof. Robert Craigen for our ﬁnal question and the accompa- nying example. Question 5. To what extent do the results in this paper generalise to weighing matrices (and complex weighing matrices and their generalisations)? In particular, is the weight of a weighing matrix a lower bound on the size of all trades in that matrix? Note that any weighing matrix has a trade of size equal to its weight, simply by negating a row. Slightly less trivially, trades with size equal to the weight can be obtained by weaving (see [4]) weighing matrices. For example, take any 2 2 block of rank one in the following W.6; 4/. The shaded entries show one such block. 0 1 00C CCC B C B 00C CC B C B 00 C B CC CC B 00 C B CC CC @ CC 00A CC 00 Trades in Complex Hadamard Matrices 221

Acknowledgements This work was inspired by the discussion after Will Orrick’s talk at the ADTHM’14 workshop, and much of the work was undertaken at the workshop. The authors are grateful to the workshop organisers and to BIRS. Research supported by ARC grants FT110100065 and DP120103067.

References

1. Billington, E.J.: Combinatorial trades: a survey of recent results. In: Wallis, W.D. (eds.) Designs 2002: Further Computational and Constructive Design Theory, pp. 47–67. Kluwer, Dordrecht (2003) 2. Bruzda, W., Tadej, W., Zyczkowski,˙ K.: Catalogue of complex Hadamard matrices. http:// chaos.if.uj.edu.pl/~karol/hadamard/ (Retrieved 10/09/2014) 3. Bryant, D., Ó Catháin, P.: An asymptotic existence result on compressed sensing matrices. Linear Algebra Appl. 475, 134–150 (2015) 4. Craigen, R.: The craft of weaving matrices. Congr. Numer. 92, 9–28 (1993) 5. Jukna, S.: Extremal Combinatorics. Texts in Theoretical Computer Science, 2nd ed. Springer, Berlin (2011) 6. Michael, T.S.: Ryser’s embedding problem for Hadamard matrices. J. Comb. Des. 14, 41–51 (2006) 7. Newman, M.: On a problem of H. J. Ryser. Linear Multilinear Algebra 12, 291–293 (1982) 8. Orrick, W.P.: Switching operations for Hadamard matrices. SIAM J. Discret. Math. 22, 31–50 (2008) 9. Petrescu, M.: Existence of continuous families of complex hadamard matrices of certain prime dimensions and related results. Ph.D. thesis, University of California, Los Angeles (1997) 10. Tadej, W., Zyczkowski,˙ K.: A concise guide to complex Hadamard matrices. Open Syst. Inf. Dyn. 13, 133–177 (2006) The Hunt for Weighing Matrices of Small Orders

Ferenc Szöllosi˝

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract In this note we use a variety of techniques to construct new weighing matrices of small orders. In particular, we construct new examples of W.n;9/ for n 2f14; 18; 19; 21g and W.n; n 1/ for n 2f42; 46g. We also discuss two possible approaches for constructing a W.66; 65/, and show nonexistence of these under certain assumptions.

Keywords Weighing matrices • Strongly regular graphs

1 Introduction

This short note is based on a talk given by the author at BIRS during the workshop “Algebraic Design Theory and Hadamard matrices” on July 14, 2014. A weighing matrix W of order n and weight w is an n n matrix with f1; 0; 1g- entries such that WWT D wI. Such matrices are denoted by W.n; w/.AW.n; n 1/ is a conference matrix, and a W.n; n/ is a Hadamard matrix [11]. A weighing matrix is symmetric if WT D W, and skew-symmetric if WT DW. Weighing matrices form an interesting class of orthogonal matrices, arising in various branches of mathematics including coding theory, design theory, and statistics. For a general treatment of weighing matrices, we refer the reader to [5, Chapter V.2.6], [6, 9]. Examples are abundant: Kharaghani and Tayfeh-Rezaie recently fully classiﬁed all Hadamard matrices of order 32: their exact number is 13:710:027 up to a natural equivalence [13].

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. F. Szöllosi˝ () Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan e-mail: [email protected]

© Springer International Publishing Switzerland 2015 223 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_19 224 F. Szöllosi˝

In this paper we use a variety of techniques to ﬁll in some gaps from the literature and construct new, interesting examples of weighing matrices. In Sect. 2 we apply numerical methods to construct symmetric or skew-symmetric W.n;9/ matrices for n 2f14; 18; 19; 21g,cf.[5, Table 2.88]. In Sect. 3 we consider symmetric conference matrices with an automorphism of order 3 and construct a new example of order 42 having full automorphism group of order 3. In Sect. 4 we use a perturbing technique to construct new matrices from old. We use Gröbner basis methods to obtain a new conference matrix of order 46 having full automorphism group of order 6,cf.[4, 17]. In Sect. 5 we investigate self-complementary strongly regular graphs of order 65 along the lines of Mathon [17], and show nonexistence of such graphs with certain properties by exhaustive computer search. However, the general case, and in particular, the existence of W.66; 65/ matrices was left open. We believe that a clever combination of these three approaches: exhaustive computer search, Gröbner basis computations, and numerical methods has high potential, and could lead to many exciting discoveries in design theory in the future.

2 Numerical Methods and Weighing Matrices of Small Orders

In this section we exhibit new weighing matrices of small orders in the spirit of Geramita and Seberry. The following philosophy is quoted verbatim from their book [9, p. 162]. In combinatorial constructions we are often motivated by the desire to fill in gaps left unresolved by other methods or to find prettier constructions. We would hope to see prettier results to cover the cases of the remainder of this section, but we are tied by a kind of a natural law: this is the way things work. One of the standard approaches in design theory to decide the existence of certain combinatorial objects is to search exhaustively for them. However, this approach is time consuming in general, and it is often the case that significant part of the search space should be explored in order to exhibit a single object. In other words, finding one example could take almost as long as finding all of them. Recent breakthroughs in design theory testify that the existence of certain designs can be concluded via probabilistic arguments [7, 12]. In essence, part of the desired combinatorial design can be chosen randomly, and intelligent modification of this random part could lead to the sought after design with positive probability. In the same spirit, instead of going through the entire search space sequentially, we hope to find a “typical” object considerably faster by numerical means. In what follows we formalize a simple approach to the search for combinatorial designs by means of numerical optimization. We remark that the idea itself is not new, see, e.g., [14]. Œ n Let S be a finite set, and assume that we seek for an S-matrix X i;jD1 D xi;j 2 S of order n satisfying some matrix equation F.X/ D 0. Then we can consider the following scalar-vector function f in n2-variables, The Hunt for Weighing Matrices 225 ! Xn Xn Y 2 2 f .x1;1; x1;2;:::;xn;n/ WD ŒF.X/i;j C .xi;j k/ ; (1) iD1 jD1 k2S and search for its global minimum. It is clear that there exist some S-matrix X satisfying F.X/ D 0 if and only if f has global minimum 0. As a warm-up result, we apply this technique to find weighing matrices W.n;9/ 14; 18; 19; 21 Œ n 1; 0; 1 of order n 2f g.Weset X i;jD1 WD xi;j,setS WD f g,and consider the scalar-vector function in formula (1) with the choice F.X/ WD XXT 9In. We have implemented this in the computer algebra system Mathematica, and by using its built-in function NMinimize[] we have established the following result after a couple of hours of computation. Theorem 2.1 (Cf. [5, pp. 290–292]). Symmetric W.n;9/exist for n 2f14; 19; 21g. There exists a skew-symmetric W.18; 9/. In particular,there are no unresolved cases of W.n;9/matrices. Proof. We set the matrix variable X in advance to be a symmetric matrix or a skew- symmetric matrix depending on the order n, and then we applied the computer search described above. The matrices found are available in Appendix 1 in explicit form. ut

A few remarks are in order. The (obvious) direct sum of weighing matrices W.n1; w/ and W.n2; w/ is a W.n1 C n2; w/, and therefore if existence follows for a range of small orders, then existence is implied for higher orders as well. From a skew- symmetric W.18; 9/ an orthogonal design of type OD.18I 1; 9/ can be constructed (see [5, Chapter V.2]), whose existence was indicated open in [9, p. 329]. There are only a few techniques to construct weighing matrices of odd orders; weaving being a notable example [6]. It would be interesting to see whether either of the symmetric W.19; 9/ or W.21; 9/ matrices have some nice feature which might be generalized to higher orders. We do not describe (the not really insightful) technical details of the implementation of our numerical approach, nevertheless we remark that the success rate of finding the global minimum of formula (1) greatly depends on the number of variables, and therefore instead of searching for the entire weighing matrix at once, it makes very much sense searching for it adaptively, row by row. The heuristic is that a “large” part of the desired weighing matrix, say the first k n rows can be chosen “almost” arbitrarily due to various underlying symmetries; and such random parts can be found very quickly by numerical means. The point is that we hope to find a suitable random part, which actually can be extended to a sought after weighing matrix, much faster this way, than by going through the search space exhaustively. The following folklore result shows that if n is even and the matrix is symmetric or skew-symmetric, then it is often enough to search for only n=2 pairwise orthogonal rows. 226 F. Szöllosi˝

ÄLemma 2.1. Let W be an orthogonal matrix of order n with block partition W D AB , where each of the blocks is of order n=2, and B is invertible. Then D D CD CAT .BT /1. Proof. By block-orthogonality, we have ACT C BDT D 0. The result follows after multiplication by B1. ut In view of Lemma 2.1 one hopes to find n=2 pairwise orthogonal rows and then, since C D BT or C DBT in the cases discussed here, D follows for free. We have briefly experimented with W.n; 16/ matrices too, but we were unable to find any of the listed open cases in the relevant tables of [5, Table 2.85]. It might be the case that none of these weighing matrices exist.

3 Conference Matrices with Fixed Point Free Automorphisms of Order 3

In this section we study symmetric conference matrices W.n; n 1/ for n Á 6 .mod 12/ having certain symmetries. The pragmatic examples of conference matrices can be obtained from the Paley graphs, but there are additional examples coming from the Belevitch construction [3], from the Seberry–Whiteman construction [19], or from self-complementary strongly regular graphs [17]. Full classiﬁcation of symmetric conference matrices is available up to order n D 34,see[21]. Assume that C is a symmetric conference matrix having a ﬁxed point free automorphism of order 3. The cycle decomposition of such automorphisms induces a block partition of C into 33 circulant blocks, which can be rearranged (via block- inversion) to the following compact form, where each of the blocks is of size n=3: 2 3 ABBT C D 4 BT AB5 : (2) BBT A

The following lemma characterizes conferences matrices of the form (2). Lemma 3.1. Let n Á 6.mod 12/. If A and B are f1; 0; 1g-matrices of order n=3, such that A D AT with zero diagonal and ˙1 entries otherwise, B is a ˙1 matrix, and furthermore the Gram equations

2 T T A C BB C B B D .n 1/In=3; (3) AB C BA C BT BT D 0; (4) hold, then the matrix C in formula (2) is a conference matrix of order n. Proof. The Eqs. (3)and(4) describe block-orthogonality. ut The Hunt for Weighing Matrices 227

Remark 3.1. The matrix equation UX C XV D W, where the matrices U; V; W,and X are all n n matrices, of which U; V,andW are given, and the problem is to ﬁnd X, is called the Sylvester equation in control theory. It is known that it has a unique solution if and only if U and V have no common eigenvalues. See [1]formore details. A simple consequence of the preceding remark is the following. Corollary 3.1. Let B be a f1; 1g-matrix of order n=3. Then there exists a unique matrix A, for which Eq. (4) holds, if and only if B and B have no common eigenvalues. Next we reduce the complexity of the system of equations (3)–(4) by invoking Lemma 2.1. We reﬁne the partition of C in formula (2) and consider its following block partition with block sizes n=6, which we rearrange as follows: 2 3 2 3 T T T T A1 A2 B1 B2 B1 B3 A1 B1 B1 A2 B2 B3 6 T T 7 6 T T 7 6 A2 B3 B2 7 6 B1 A1 B1 B3 A2 B2 7 6 T T 7 6 T T 7 6 B B A1 A2 B1 B2 7 6 B1 B A1 B2 B A2 7 C D 6 1 3 7 6 1 3 7 : (5) 6 T T 7 6 T T 7 6 B2 A2 B3 7 6 A2 B3 B2 7 4 T T 5 4 T T 5 B1 B2 B1 B3 A1 A2 B2 A2 B3 T T T T B3 B2 A2 B3 B2 A2

Proposition 3.1. Let n Á 6.mod 12/.IfA1; A2; B1; B2, and B3 are f1; 0; 1g- T matrices of order n=6, such that A1 D A1 with zero diagonal and ˙1 entries otherwise, A2; B1; B2, and B3 are ˙1 matrices, and furthermore the Gram equations

2 T T T T T A1 C B1B1 C B1 B1 C A2A2 C B2B2 C B3 B3 D .n 1/In=6; (6) T T T T T A1B1 C B1A1 C B1 B1 C A2B3 C B2A2 C B3 B2 D 0; (7) hold, then the ﬁrst three block of rows of the matrix C in formula (5) .on the right/ are pairwise orthogonal. If, in addition, the block matrix 2 3 T A2 B2 B3 4 T 5 B3 A2 B2 T B2 B3 A2 is invertible, then the lower right part of C follows uniquely. Proof. The Eqs. (6)and(7) describe block-orthogonality. The lower right part of C follows uniquely if the conditions of Lemma 2.1 are met. ut An adaptive implementation of the numerical methods described in Sect. 2 leads to the following result. 228 F. Szöllosi˝

Theorem 3.1. There exist symmetric conference matrices of the form (2) for n 2 f6; 18; 30; 42g. Proof. We have found solutions to Eqs. (6)–(7) of Proposition 3.1. The cases n 2 f6; 18; 30g are quite easy to obtain, see, e.g., the online repository [20]. Therefore we only present the matrix B of size 14 corresponding to the case n D 42 in Appendix 2. Since B and its negative do not have a common eigenvalue, by Corollary 3.1 the matrix A follows uniquely by solving the linear system (4). This W.42; 41/ matrix has full automorphism group of order 3, and therefore it is inequivalent from any of the 18 known examples [4], [5, Table 13.62], [20]. ut The question of whether a conference matrix with block form (2) exists for n 2 f54; 66g remains open. We remark that while the case n D 54 might be doable, the case n D 66 seems quite difﬁcult to handle with conventional methods (i.e., with exhaustive computer search); see more on this in [2]. On the other hand, the case n D 78 is impossible, since n 1 D 77 is not a sum of two square of integers [15], [18].

4 New Matrices from Old

In this section we describe a “perturbing” technique and obtain a new conference matrix order 46 having full automorphism group of order 6. We remark that since 45 is not a prime power, it is entirely non-trivial to exhibit such a conference matrix in the first place. Mathon, via an ingenious construction—as Seberry and Whiteman call it in [19, p. 365]—obtained such matrices first in [16], and then found additional examples in [4, 17]. Having constructed the incidence matrix D of a combinatorial design, it is often desirable to construct additional examples by various “switching” techniques. This usually amounts to modifying some parts of the design in order to escape equivalence classes. Here we explore the possibility of switching arbitrary parts, chosen randomly. The idea is to start with an m nS-matrix D, describing some combinatorial object via the matrix equation F.D/ D 0, and then replace some part of it with unknown variables xi, 1 Ä i Ä s mn to obtain a parametrized matrix X WD D.x1;:::;xs/. The goal is now to find all s-tuples .x1;:::;xs/ for which the matrix X has the desired properties, which are usually the same as the properties of D. We attemptP to solveQ the matrix equation F.X/ D 0, satisfying the additional s . /2 0 constraints that iD1 k2S xi k D , hoping to discover inequivalent solutions to D. A nice feature of this technique is that some solutions are always guaranteed (including the matrix D itself). Assume that C is a symmetric conference matrix of order n, partitioned into blocks of order n=2 as follows: " # AB C D 1 ; (8) BT BT A BT The Hunt for Weighing Matrices 229 where we assume that the rows and columns of C could have been rearranged in advanceinawaysothatB is invertible. Then we replace the blocks A and B with 2 A.x/ WD A.x1;:::;xu/ and B.y/ WD B.xuC1;:::;xs/,wheres n =2 distinct positions were replaced by unknown variables, maintaining A.x/ D A.x/T and ŒA.x/i;i D 0 for 1 Ä i Ä n=2. Then, we attempt to solve the matrix equation

2 T .A.x// C B.y/B.y/ .n 1/In=2 D 0; (9)

2 1 0 1 . / 0 with respect to xi D , Ä i Ä s, and detB y ¤ , e.g. by computing a Gröbner basis [8]. For every solutions .x; y/ we reconstruct C via formula (8). We remark that nothing guarantees that the lower right block of C.x; y/ is a f1; 0; 1g-matrix, and therefore this should be analyzed later. We have applied this technique to a conference matrix of order 46, taken from [20], and found a new matrix. Theorem 4.1. There exists a new W.46; 45/, having full automorphism group of order 6. Proof. We took a matrix from [20] and replaced s D 236 entries in its blocks A and B with unknown variables xi, 1 Ä i Ä s. Then we have solved Eq. (9) subject to 2 1 0 1 xi D for Ä i Ä s by computing a Gröbner basis with the computer algebra system magma. The matrices A.x/ and B.y/, along with the computed Gröbner basis are available in Appendix 3. It turns out that exactly 12 distinct matrices can be found, and one of these has full automorphism group of order 6. In comparison, all previously known matrices of this size have an automorphism group of order 2, 3, 8, 10,or24,see[4, 17, 20]. ut

5 On Self-Complementary Strongly Regular Graphs on 65 Vertices

Mathon in a seminal paper described a computational approach towards self- complementary strongly regular graphs [17], which are essentially symmetric conference matrices [17, Theorem 2]. He was able to construct such graphs on n D 45 vertices with higher degree of symmetry than previously [4, 16]. Mathon’s idea was to consider an automorphism induced by a complementing permutation of a strongly regular graph, then classify the possible block-valency (or orbit-) matrices, and finally search for suitable circulant matrices with the given valencies. We believe that his approach is probably the most promising way towards constructing a W.66; 65/—if such a matrix exists. For terminology and details, we refer the reader to [17]. Let A be a symmetric matrix which is partitioned into s2 rectangular blocks, where each block has constant row and column sum. Let R be the corresponding 230 F. Szöllosi˝ block-valency matrix, that is, Ri;j equals to the row sum of the .i; j/th block of A.It turns out that if A satisfies a quadratic matrix equation, then so does R. Proposition 5.1 (See [2], [5, Chapter VII.6.3], [17]). Assume that A is a symmetric matrix which is partitioned into s2 rectangular blocks, where each block has constant row and column sum. Let R denote the corresponding block valency matrix of order s, let r be the column vector containing the sizes of the .square/ diagonal blocks, and let WD diag.r/. Then, if A2 D ˛I C ˇJ C Aforsome˛, ˇ, and , then RT R D ˛ C ˇrrT C R. The point is that the size of the matrix R can be considerably smaller than of the size of A, and solving the quadratic equation for R could be much simpler. Proof. For any 1 Ä x Ä s let .x/ be the characteristic vector of the xth block of columns of the matrix A,thatis,.x/k D 1 if and only if the kth column of A constitute its xth block of columns, and .x/k D 0 otherwise. To get the .i; j/th entry of the matrices of the claimed equation above, multiply the matrix equation A2 D ˛I C ˇJ C A from the left and the right by .i/ and .j/T , respectively. ut Mathon succeeded in constructing self-complementary strongly regular graphs of order n D 45 having non-uniform -partitions, and one might hope that such phenomenon occurs for other non-prime power orders n too. Therefore we investigated some of the possible -partitions of such graphs on n D 65 vertices. The case when has three orbits of size 16 and two orbits of size 8 seemed very promising at first, since we were able to classify all block-valency matrices, see them in Appendix 4. Unfortunately, none of these lead to W.66; 65/ matrices. Theorem 5.1. There does not exist any self-complementary strongly regular graphs on n D 65 vertices with complementing permutation of type .4;4;4;3;3/. Proof. We classified all block-valency matrices of this type by Proposition 5.1: three such matrices were found up to rearranging the blocks. These matrices are listed in Appendix 4. Having classified all block-valency matrices, we performed an exhaustive computer search for the block-circulant matrices with the given block valencies. Unfortunately, our efforts were fruitless, as it was impossible to find 33 rows of a conference graph of this type. ut It is very likely that other type of block valency matrices are to be found; the case where all 16 blocks are of size 4 being the most probable one. We believe that with sufficient computing power the block-valency matrices can be classified. Those corresponding to an automorphism of order 5 have already been determined in [2]. Therefore the search for weighing matrices continues. We conclude the paper with a message from Horadam [11, p. 237]: “Good fortune to those hunting for solutions.”

Acknowledgements We are greatly indebted to Prof. A. Munemasa who pointed out that certain maximal self-orthogonal ternary codes of length 19 (see [10]) contain symmetric W.19; 9/ matrices, and provided us with such an example. The Hunt for Weighing Matrices 231

This work was supported in part by the JSPS KAKENHI Grant Numbers 24 02807.Part of the computational results in this research were obtained using supercomputing resources at Cyberscience Center, Tohoku University.

Appendix 1: New Weighing Matrices of Orders n 2f14; 18; 19; 21g

W14:={{0,0,0,1,0,-1,-1,1,1,0,-1,1,1,1}, {0,0,-1,1,0,-1,0,-1,-1,0,1,-1,1,1}, {0,-1,0,1,0,-1,0,1,-1,0,-1,-1,-1,-1}, {1,1,1,0,-1,-1,1,0,1,1,0,-1,0,0}, {0,0,0,-1,0,-1,0,-1,-1,1,-1,1,1,-1}, {-1,-1,-1,-1,-1,0,-1,0,1,1,0,-1,0,0}, {-1,0,0,1,0,-1,0,-1,1,0,1,1,-1,-1}, {1,-1,1,0,-1,0,-1,0,0,-1,1,0,1,-1}, {1,-1,-1,1,-1,1,1,0,0,1,0,1,0,0}, {0,0,0,1,1,1,0,-1,1,0,-1,-1,1,-1}, {-1,1,-1,0,-1,0,1,1,0,-1,0,0,1,-1}, {1,-1,-1,-1,1,-1,1,0,1,-1,0,0,0,0}, {1,1,-1,0,1,0,-1,1,0,1,1,0,0,-1}, {1,1,-1,0,-1,0,-1,-1,0,-1,-1,0,-1,0}};

W18:={{0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}, {0,0,-1,0,0,0,0,1,-1,0,1,-1,1,-1,0,-1,1,0}, {0,1,0,0,-1,-1,1,-1,0,0,1,0,0,1,-1,-1,0,0}, {0,0,0,0,0,-1,-1,0,1,-1,0,1,1,0,1,-1,0,-1}, {0,0,1,0,0,-1,-1,-1,-1,0,0,0,1,-1,-1,1,0,0}, {0,0,1,1,1,0,1,0,1,-1,1,0,0,-1,0,0,0,1}, {0,0,-1,1,1,-1,0,-1,0,1,0,-1,0,0,1,0,-1,0}, {0,-1,1,0,1,0,1,0,-1,1,0,1,0,0,0,-1,0,-1}, {0,1,0,-1,1,-1,0,1,0,0,1,0,-1,0,0,1,0,-1}, {-1,0,0,1,0,1,-1,-1,0,0,1,0,-1,0,0,0,1,-1}, {-1,-1,-1,0,0,-1,0,0,-1,-1,0,1,-1,0,0,0,0,1}, {-1,1,0,-1,0,0,1,-1,0,0,-1,0,0,-1,1,0,1,0}, {-1,-1,0,-1,-1,0,0,0,1,1,1,0,0,-1,0,0,-1,0}, {-1,1,-1,0,1,1,0,0,0,0,0,1,1,0,-1,0,-1,0}, {-1,0,1,-1,1,0,-1,0,0,0,0,-1,0,1,0,-1,0,1}, {-1,1,1,1,-1,0,0,1,-1,0,0,0,0,0,1,0,-1,0}, {-1,-1,0,0,0,0,1,0,0,-1,0,-1,1,1,0,1,0,-1}, {-1,0,0,1,0,-1,0,1,1,1,-1,0,0,0,-1,0,1,0}};

W19:={{0,-1,0,1,1,1,1,0,-1,0,0,0,-1,0,0,1,0,1,0}, {-1,1,0,0,0,0,0,-1,-1,-1,0,0,0,1,-1,0,-1,0,1}, {0,0,0,-1,1,-1,-1,0,0,0,0,1,0,0,1,1,0,1,1}, {1,0,-1,0,0,0,0,1,-1,-1,0,1,1,-1,-1,0,0,0,0}, {1,0,1,0,1,1,-1,1,0,0,0,0,0,1,0,0,-1,-1,0}, {1,0,-1,0,1,0,0,-1,1,-1,-1,0,-1,0,0,-1,0,0,0}, {1,0,-1,0,-1,0,0,0,0,-1,1,-1,0,1,1,1,0,0,0}, {0,-1,0,1,1,-1,0,0,0,0,1,-1,1,0,0,-1,0,0,1}, {-1,-1,0,-1,0,1,0,0,1,-1,1,0,0,-1,0,0,-1,0,0 {0,-1,0,-1,0,-1,-1,0,-1,0,0,-1,-1,0,-1,0,0,0,-1}, {0,0,0,0,0,-1,1,1,1,0,0,0,-1,0,-1,1,0,-1,1}, {0,0,1,1,0,0,-1,-1,0,-1,0,0,0,-1,0,1,1,-1,0}, {-1,0,0,1,0,-1,0,1,0,-1,-1,0,0,0,1,0,-1,0,-1}, {0,1,0,-1,1,0,1,0,-1,0,0,-1,0,-1,1,0,0,-1,0}, {0,-1,1,-1,0,0,1,0,0,-1,-1,0,1,1,0,0,1,0,0}, {1,0,1,0,0,-1,1,-1,0,0,1,1,0,0,0,0,-1,0,-1}, {0,-1,0,0,-1,0,0,0,-1,0,0,1,-1,0,1,-1,0,-1,1}, {1,0,1,0,-1,0,0,0,0,0,-1,-1,0,-1,0,0,-1,1,1}, {0,1,1,0,0,0,0,1,0,-1,1,0,-1,0,0,-1,1,1,0}};

W21:={{0,0,1,1,-1,0,0,0,0,1,1,1,0,-1,0,0,1,0,1,0,0}, {0,0,0,-1,-1,0,-1,-1,0,1,-1,0,0,0,0,0,0,1,0,-1,-1}, {1,0,0,0,1,0,1,-1,1,0,-1,0,-1,0,0,0,1,0,1,0,0}, {1,-1,0,0,-1,1,0,-1,0,-1,0,0,0,-1,0,0,0,0,-1,0,1}, {-1,-1,1,-1,0,0,1,0,0,0,0,0,0,0,1,1,0,1,0,1,0}, {0,0,0,1,0,1,1,-1,-1,1,0,-1,1,1,0,0,0,0,0,0,0}, {0,-1,1,0,1,1,0,0,0,0,0,1,0,0,0,0,-1,-1,0,-1,-1}, {0,-1,-1,-1,0,-1,0,-1,0,0,1,0,1,0,0,0,0,-1,1,0,0}, {0,0,1,0,0,-1,0,0,-1,0,-1,-1,0,-1,-1,1,0,-1,0,0,0}, {1,1,0,-1,0,1,0,0,0,0,1,0,0,0,-1,1,0,0,0,1,-1}, {1,-1,-1,0,0,0,0,1,-1,1,-1,1,0,0,0,0,0,0,0,1,0}, {1,0,0,0,0,-1,1,0,-1,0,1,0,-1,0,0,0,-1,1,0,-1,0}, {0,0,-1,0,0,1,0,1,0,0,0,-1,0,-1,1,1,0,0,1,-1,0}, {-1,0,0,-1,0,1,0,0,-1,0,0,0,-1,0,-1,-1,0,0,1,0,1}, {0,0,0,0,1,0,0,0,-1,-1,0,0,1,-1,0,-1,1,1,0,0,-1}, {0,0,0,0,1,0,0,0,1,1,0,0,1,-1,-1,0,-1,1,0,0,1}, {1,0,1,0,0,0,-1,0,0,0,0,-1,0,0,1,-1,-1,0,1,1,0}, {0,1,0,0,1,0,-1,-1,-1,0,0,1,0,0,1,1,0,0,0,0,1}, {1,0,1,-1,0,0,0,1,0,0,0,0,1,1,0,0,1,0,0,-1,1}, {0,-1,0,0,1,0,-1,0,0,1,1,-1,-1,0,0,0,1,0,-1,0,0}, {0,-1,0,1,0,0,-1,0,0,-1,0,0,0,1,-1,1,0,1,1,0,0}};

Appendix 2: A New Conference Matrix of Order 42

We only present the matrix B leading to a W.42; 41/ via formulae (2)and(4).

B:={{1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1}, {-1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1}, {-1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, -1, 1, -1}, {-1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, -1}, {-1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1}, {1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1}, {-1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1}, {-1, -1, 1, -1, 1, 1, 1, 1, 1, -1, -1, 1, -1, -1}, {1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, 1}, {1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, -1, 1, 1}, {1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1}, {1, 1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, -1}, {-1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1}, {-1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, -1, 1}}; 232 F. Szöllosi˝

Appendix 3: A New Conference Matrix of Order 46

Here we present the matrices A.x/ and B.y/ leading to 12 distinct W.46; 45/ matrices via formula (8).

Ax:={{0,1,-1,1,1,-1,1,-1,1,-1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,1}, {1,0,-1,-1,-1,1,1,-1,1,-1,-1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1}, {-1,-1,0,1,-1,1,1,-1,-1,1,-1,-1,-1,-1,1,1,-1,-1,-1,1,1,1,1}, {1,-1,1,0,1,1,-1,-1,-1,1,-1,-1,-1,-1,-1,-1,1,1,1,-1,1,1,1}, {1,-1,-1,1,0,1,-1,1,-1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,-1,1,-1}, {-1,1,1,1,1,0,-1,-1,1,1,1,-1,-1,1,-1,1,-1,1,1,1,1,-1,-1}, {1,1,1,-1,-1,-1,0,1,-1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,-1,-1}, {-1,-1,-1,-1,1,-1,1,0,1,-1,-1,1,-1,1,1,1,1,-1,-1,-1,1,-1,-1}, {1,1,-1,-1,-1,1,-1,1,0,1,-1,-1,1,1,1,-1,-1,1,-1,-1,1,-1,-1}, {-1,-1,1,1,-1,1,-1,-1,1,0,1,1,-1,1,1,x[1],x[14],x[27],x[40],x[53],x[66],x[79],x[92]}, {-1,-1,-1,-1,1,1,1,-1,-1,1,0,1,1,1,1,x[2],x[15],x[28],x[41],x[54],x[67],x[80],x[93]}, {1,1,-1,-1,1,-1,1,1,-1,1,1,0,1,-1,-1,x[3],x[16],x[29],x[42],x[55],x[68],x[81],x[94]}, {-1,-1,-1,-1,-1,-1,1,-1,1,-1,1,1,0,1,-1,x[4],x[17],x[30],x[43],x[56],x[69],x[82],x[95]}, {-1,-1,-1,-1,1,1,-1,1,1,1,1,-1,1,0,1,x[5],x[18],x[31],x[44],x[57],x[70],x[83],x[96]}, {1,-1,1,-1,-1,-1,1,1,1,1,1,-1,-1,1,0,x[6],x[19],x[32],x[45],x[58],x[71],x[84],x[97]}, {1,-1,1,-1,1,1,-1,1,-1,x[1],x[2],x[3],x[4],x[5],x[6],0,x[20],x[33],x[46],x[59],x[72],x[85],x[98]}, {-1,1,-1,1,-1,-1,1,1,-1,x[14],x[15],x[16],x[17],x[18],x[19],x[20],0,x[34],x[47],x[60],x[73],x[86], x[99]}, {-1,1,-1,1,1,1,-1,-1,1,x[27],x[28],x[29],x[30],x[31],x[32],x[33],x[34],0,x[48],x[61],x[74],x[87], x[100]}, {1,1,-1,1,-1,1,1,-1,-1,x[40],x[41],x[42],x[43],x[44],x[45],x[46],x[47],x[48],0,x[62],x[75],x[88], x[101]}, {-1,1,1,-1,-1,1,1,-1,-1,x[53],x[54],x[55],x[56],x[57],x[58],x[59],x[60],x[61],x[62],0,x[76],x[89], x[102]}, {1,-1,1,1,-1,1,1,1,1,x[66],x[67],x[68],x[69],x[70],x[71],x[72],x[73],x[74],x[75],x[76],0,x[90], x[103]}, {1,-1,1,1,1,-1,-1,-1,-1,x[79],x[80],x[81],x[82],x[83],x[84],x[85],x[86],x[87],x[88],x[89],x[90],0, x[104]}, {1,-1,1,1,-1,-1,-1,-1,-1,x[92],x[93],x[94],x[95],x[96],x[97],x[98],x[99],x[100],x[101],x[102],x[103], x[104],0}};

By:={{1,1,-1,-1,-1,1,1,-1,1,1,-1,1,-1,-1,-1,1,-1,1,-1,-1,1,1,-1}, {1,1,1,1,-1,-1,1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,1,1,1,1,1}, {1,1,1,-1,1,-1,1,1,-1,1,1,-1,1,1,-1,-1,-1,1,-1,1,-1,-1,1}, {1,-1,1,-1,-1,1,-1,1,1,-1,1,1,1,1,-1,-1,-1,-1,1,-1,1,1,-1}, {1,1,1,1,1,-1,1,-1,1,-1,-1,1,1,1,-1,1,-1,-1,-1,1,1,-1,-1}, {1,1,-1,1,-1,1,1,1,-1,-1,1,-1,1,-1,1,1,-1,-1,-1,-1,-1,-1,-1}, {1,-1,1,-1,1,1,1,1,1,1,-1,-1,1,1,-1,1,-1,-1,-1,-1,-1,-1,-1}, {1,-1,-1,1,1,1,1,1,-1,-1,1,1,1,1,-1,-1,1,-1,1,-1,1,-1,1}, {1,1,-1,-1,-1,-1,1,1,1,1,-1,1,1,-1,1,-1,-1,1,1,-1,1,-1,1}, {1,-1,1,-1,-1,-1,1,1,1,1,-1,1,-1,-1,1,x[105],x[133],x[161],x[189],x[217],x[245],x[273],x[301]}, {1,-1,-1,1,1,1,-1,-1,1,1,1,-1,1,-1,-1,x[107],x[135],x[163],x[191],x[219],x[247],x[275],x[303]}, {1,1,-1,-1,-1,-1,1,1,-1,-1,1,1,-1,1,-1,x[109],x[137],x[165],x[193],x[221],x[249],x[277],x[305]}, {1,1,1,-1,-1,1,-1,1,1,-1,-1,-1,1,1,1,x[111],x[139],x[167],x[195],x[223],x[251],x[279],x[307]}, {1,1,1,-1,-1,1,1,-1,-1,-1,-1,-1,-1,1,-1,x[113],x[141],x[169],x[197],x[225],x[253],x[281],x[309]}, {1,-1,-1,1,-1,-1,-1,1,1,-1,-1,-1,-1,1,-1,x[115],x[143],x[171],x[199],x[227],x[255],x[283],x[311]}, {1,-1,-1,1,-1,-1,1,-1,-1,x[106],x[108],x[110],x[112],x[114],x[116],x[118],x[145],x[173],x[201],x[229], x[257],x[285],x[313]}, {1,-1,-1,-1,1,-1,-1,1,1,x[134],x[136],x[138],x[140],x[142],x[144],x[146],x[148],x[175],x[203],x[231], x[259],x[287],x[315]}, {1,-1,-1,-1,1,-1,1,-1,-1,x[162],x[164],x[166],x[168],x[170],x[172],x[174],x[176],x[178],x[205],x[233], x[261],x[289],x[317]}, {1,-1,1,1,-1,1,-1,-1,-1,x[190],x[192],x[194],x[196],x[198],x[200],x[202],x[204],x[206],x[208],x[235], x[263],x[291],x[319]}, {1,1,1,1,1,-1,-1,-1,-1,x[218],x[220],x[222],x[224],x[226],x[228],x[230],x[232],x[234],x[236],x[238], x[265],x[293],x[321]}, {1,1,-1,-1,1,1,-1,-1,-1,x[246],x[248],x[250],x[252],x[254],x[256],x[258],x[260],x[262],x[264],x[266], x[268],x[295],x[323]}, {1,1,-1,-1,1,1,-1,1,-1,x[274],x[276],x[278],x[280],x[282],x[284],x[286],x[288],x[290],x[292],x[294], x[296],x[298],x[325]}, {1,1,-1,-1,1,1,1,-1,1,x[302],x[304],x[306],x[308],x[310],x[312],x[314],x[316],x[318],x[320],x[322], x[324],x[326],x[328]}}; The following is the relevant Gröbner basis:

GB:={-1+x[1],-1+x[98]+x[99]+x[100],1+x[101],-1+x[102],1+x[103],1+x[104],1+x[105],1+x[106],-1+x[107], 1+x[108],1+x[109],1+x[110],1+x[111],-1+x[112],-1+x[113],1+x[114],-1+x[115],-1+x[116],-1+x[118], -1+x[133],1+x[134],1+x[135],1+x[136],-1+x[137],1+x[138],-1+x[139],-1+x[14],-1+x[140],-1+x[141], 1+x[142],1+x[143],-1+x[144],-1+x[145],-1+x[146],-1+x[148],1+x[15],-1+x[16],1+x[161],1+x[162], -1+x[163],1+x[164],-1+x[165],1+x[166],-1+x[167],-1+x[168],1+x[169],1+x[17],-1+x[170],-1+x[171], -1+x[172],-1+x[173],1+x[174],1+x[175],1+x[176],-1+x[178],1+x[18],-1+x[189],1+x[19],1+x[190], -1+x[191],1+x[192],1+x[193],-1+x[194],1+x[195],1+x[196],-1+x[197],x[94]+x[198],-1+x[199],1+x[2], -1+x[20],x[94]+x[200],1+x[201],x[94]+x[202],-1+x[203],-x[87]+x[204],-1+x[205],-x[86]+x[206], -1+x[86]+x[87]+x[208],-x[94]+x[217],1+x[218],-x[94]+x[219],1+x[220],-x[94]+x[221],-1+x[222], -x[86]-x[87]+x[98]+x[223],1+x[224],-1+x[86]+x[99]+x[225],-x[94]+x[226],x[87]-x[98]-x[99]+x[227], -x[94]+x[228],-x[86]-x[87]+x[98]+x[229],-x[94]+x[230],-1+x[86]+x[99]+x[231],-1+x[98]+x[99]+x[232], x[87]-x[98]-x[99]+x[233],-x[99]+x[234],1+x[235],-x[98]+x[236],1+x[238],1+x[245],1+x[246],1+x[247], 1+x[248],1+x[249],-1+x[250],-x[98]+x[251],-1+x[252],-x[99]+x[253],1+x[254],-1+x[98]+x[99]+x[255], 1+x[256],-x[98]+x[257],1+x[258],-x[99]+x[259],x[87]-x[98]-x[99]+x[260],-1+x[98]+x[99]+x[261], -1+x[86]+x[99]+x[262],1+x[263],-x[86]-x[87]+x[98]+x[264],-1+x[265],-1+x[266],1+x[268],1+x[27], The Hunt for Weighing Matrices 233

x[94]+x[273],-1+x[274],x[94]+x[275],1+x[276],x[94]+x[277],1+x[278],-x[86]-x[87]+x[98]+x[279], -1+x[28],1+x[280],-1+x[86]+x[99]+x[281],-x[94]+x[282],x[87]-x[98]-x[99]+x[283],-x[94]+x[284], -x[86]-x[87]+x[98]+x[285],-x[94]+x[286],-1+x[86]+x[99]+x[287],-x[87]+x[288],x[87]-x[98]-x[99]+x[289], -1+x[29],-x[86]+x[290],1+x[291],-1+x[86]+x[87]+x[292],1+x[293],-1+x[294],-1+x[295],-1+x[296], 1+x[298],-1+x[3],-1+x[30],1+x[301],1+x[302],1+x[303],-1+x[304],1+x[305],1+x[306], -1+x[86]+x[87]+x[307],1+x[308],-x[86]+x[309],1+x[31],x[94]+x[310],-x[87]+x[311],x[94]+x[312], -1+x[86]+x[87]+x[313],x[94]+x[314],-x[86]+x[315],-1+x[98]+x[99]+x[316],-x[87]+x[317],-x[99]+x[318], -1+x[319],1+x[32],-x[98]+x[320],1+x[321],1+x[322],1+x[323],-1+x[324],-1+x[325],-1+x[326],-1+x[328], 1+x[33],-1+x[34],1+x[4],x[40]-x[94],x[41]-x[94],x[42]-x[94],-1+x[43]+x[86]+x[87],x[44]-x[86], x[45]-x[87],-1+x[46]+x[86]+x[87],x[47]-x[86],x[48]-x[87],1+x[5],x[53]+x[94],x[54]+x[94],x[55]+x[94], x[56]-x[98],x[57]-x[99],-1+x[58]+x[98]+x[99],x[59]-x[98],-1+x[6],x[60]-x[99],-1+x[61]+x[98]+x[99], -1+x[62],1+x[66],1+x[67],1+x[68],x[69]-x[86]-x[87]+x[98],-1+x[70]+x[86]+x[99],x[71]+x[87]-x[98]-x[99], x[72]-x[86]-x[87]+x[98],-1+x[73]+x[86]+x[99],x[74]+x[87]-x[98]-x[99],-1+x[75],-1+x[76],x[79]+x[94], x[80]+x[94],x[81]+x[94],-1+x[82]+x[86]+x[87],x[83]-x[86],x[84]-x[87],-1+x[85]+x[86]+x[87],-1+x[86]^2, 1-x[86]-x[87]+x[86]*x[87],-x[86]-x[87]+x[98]+x[86]*x[98]+x[99]-x[87]*x[99],1-x[86]-x[99]+x[86]*x[99], -1+x[87]^2,-x[98]+x[87]*x[98]-x[99]+x[87]*x[99],-1+x[88],1+x[89],1+x[90],x[92]-x[94],x[93]-x[94], -1+x[94]^2,x[95]-x[98],x[96]-x[99],-1+x[97]+x[98]+x[99],-1+x[98]^2,1-x[98]-x[99]+x[98]*x[99], -1+x[99]^2}; The following solution vector .xŒ1; xŒ2; : : : ; xŒ328/ leads to a W.46; 45/ with full automorphism group of order 6.

s:={1,-1,1,-1,-1,1,1,-1,1,-1,-1,-1,1,-1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,1,-1,1,1,1,1,1,1,1,-1,1,1,-1,1, -1,-1,-1,1,-1,1,1,-1,1,1,1,1,1,1,-1,1,1,-1,1,1,1,-1,-1,-1,-1,-1,1,1,-1,1,1,-1,-1,1,-1,-1,-1,-1,1, -1,-1,-1,-1,1,1,-1,1,1,1,1,-1,-1,-1,1,-1,1,1,1,-1,-1,1,1,1,1,-1,-1,1,-1,1,-1,1,1,-1,1,1,1,1,-1,-1, -1,1,1,-1,1,-1,-1,1,-1,-1,1,1,1,1,-1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,1,1,-1,-1,-1,1,-1,1,-1,-1,-1,1,1, -1,1,-1,-1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,1,-1,1,1,-1,-1,-1,1,1,1,-1,1,1,1,-1,1,-1,1,-1,-1,-1,1,-1,1, -1,-1,1,1,1,-1,-1,-1,1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,1,1,1,1,-1,1,1,-1,1,1,1,1,-1,-1,-1,1,1,1,1};

Appendix 4: Block Valency Matrices for n D 65

The following block valency matrices were found for the complementing permutations of type .4;4;4;3;3/:

2 3 2 3 2 3 0 808080 4040 0 808080 4040 0 808080 4040 6 7 6 7 6 7 6 1 642444 12227 6 1 344244 22247 6 1 444345 02327 6 7 6 7 6 7 6 0 414644 23227 6 0 446444 22027 6 0 435434 24217 6 7 6 7 6 7 6 7 6 7 6 7 6 1 243444 22427 6 1 463442 23217 6 1 455432 22137 6 7 6 7 6 7 6 0 464444 22207 6 0 244464 12327 6 0 344265 22137 6 7 6 7 6 7 6 7 6 7 6 7 R1 D 6 1 444424 42127 ; R2 D 6 1 444634 20227 ; R3 D 6 1 433654 22117 : 6 7 6 7 6 7 6 0 444445 20237 6 0 442444 42227 6 0 542542 22337 6 7 6 7 6 7 6 7 6 7 6 7 6 1 244484 12027 6 1 444248 12207 6 1 044444 32427 6 7 6 7 6 7 6 0 464440 22247 6 0 446404 22427 6 0 484444 20207 6 7 6 7 6 7 4 1 448424 02125 4 1 404644 24125 4 1 642226 42125 0 444046 2422 0 842444 0222 0 426626 2022

References

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Mieko Yamada

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Abstract We know there exists a family of Menon–Hadamard difference sets over Galois rings of characteristic of an even power of 2 and of an odd extension degree, which has a nested structure. The projective limit of these Menon–Hadamard difference sets is a non-empty subset of a valuation ring of a local field. Conversely, does there exist a subset of a local field whose image by the natural projection always gives a difference set over a Galois ring? We will show an answer to this problem. A family of Menon–Hadamard difference sets is obtained from a subgroup of a valuation ring of a local field by the natural projections and it also has a nested structure. The formal group and the p-adic logarithm function serve an important role to the construction.

Keywords Galois ring • Menon-Hadamard difference set • Local ﬁeld • p-adic • Logaritm function • Formal group

1 Introduction

Difference sets with the parameters v D 22n; k D 2n1.2n 1/; D 2n1.2n1 1/ are well known and have been studied over several kinds of algebraic structures. They are called Menon–Hadamard difference sets (see, e.g., [6, 7]). Kraemer ﬁnally proved that Turyn’s exponent bound is necessary and sufﬁcient conditions of existence of a Menon–Hadamard difference set [9]. We showed that there exists a family of Menon–Hadamard difference sets over Galois rings of characteristic of an ever power of 2 and of an odd extension degree [16]. Though this family produces no new orders, it has an interesting property, a nested structure. That is,

This paper is in ﬁnal form and no similar paper has been or is being submitted elsewhere. M. Yamada () School of Arts and Sciences, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan e-mail: [email protected]

© Springer International Publishing Switzerland 2015 235 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_20 236 M. Yamada the difference set over the Galois ring of characteristic 2n is embedded in the ideal part of a difference set over the Galois ring of characteristic 2nC2 for every even n. The projective limit of Galois rings is a valuation ring of a local field (see, e.g., [4, 8]). Then the projective limit of these Menon–Hadamard difference sets is a non- empty subset of a valuation ring of a local field. Thus the following question arises: Does there exist a subset of a local field whose image by the natural projection always gives a difference set over a Galois ring? We give an answer to this question. 2n pl 2n For a characteristic and an ideal n of a Galois ring of characteristic ,we define the subgroup W of a valuation ring of a local field. Then we construct a Menon–Hadamard difference set over a Galois ring from W by national projections. Then by fixing the extension degree and varying n and l, we obtain a family of Menon–Hadamard difference sets. The formal group and the p-adic logarithm function (see, e.g., [4, 8, 13]) serve an important role to the construction. There have been several results of relative difference sets and partial difference sets using Galois rings [2, 5, 15]. Polhill showed the construction of nested partial difference sets [14]. The property such that the difference set over a group (or a field or a ring) implies the difference set over a subgroup (or a subfield or a subring) will be called a nested property. If every difference set of a family of difference sets is nested, we will say a family has a nested structure. The family by our construction has a nested structure, that is, 2Dn DnC2 where Dn is the difference set over the Galois ring of n characteristic 2 and DnC2 is the difference set over the Galois ring of characteristic 2nC2. Many authors have given the conditions of the groups in which the Hadamard difference set has a nested property (see Davis and Jedwab’s survey [3]). As Menon’s direct product construction of difference sets, there are theorems that a larger difference set is constructed from some smaller difference sets (see, e.g., [11]). A family of difference sets is obtained by applying the methods repeatedly starting from existing difference sets, which is called a recursive construction. Our construction is different from a recursive construction. The subgroup W of a valuation ring used in our construction is infinite and not a difference set. Our difference set is constructed from the images of the infinite subsets by natural projections. Hence it does not need the assumption of existence of a difference set. To my knowledge, there are a few results on p-adic codes (Calderbank and Sloane [1] and Lagorce [10]), but the results of p-adic difference sets are not known. It is difficult to define combinatorial concepts over infinite algebraic structures extending them defined over finite algebraic structure. We think the first step is to find some relations between combinatorial concepts and some concepts over infinite algebraic structures.

2 Galois Rings

n Let Z be a rational integer ring and denote Z=2 Z by An. A monic irreducible polynomial '.x/ 2 AnŒx of degree s is called a basic primitive polynomial if the image of '.x/ under the natural projection is a primitive polynomial over GF.2/. Menon–Hadamard Difference Sets Obtained from a Local Field 237

Let '.x/ 2 AnŒx be a basic primitive polynomial of degree s and denote the root of '.x/ by .ThenAnŒx='.x/ is a Galois extension of An and is called a Galois ring of characteristic 2n and of an extension degree s, denoted by GR.2n; s/.The extension ring of An obtained by adjoining is isomorphic to AnŒx='.x/. For easy n reference, we put Rn D GR.2 ; s/. Rn is a local ring and has a unique maximal p 2R R pl 2lR ;0 ideal n D n. Every ideal of n is n D n Ä l Ä n. The residue class s field Rn=pn is isomorphic to a finite field GF.2 /. We take the Teichmüller system 2s2 Tn Df0; 1; ; ; g as a set of complete representatives of Rn=pn. The additive group of GR.2n; s/ is an abelian group .Z=2nZ/s. An arbitrary element of ˛ of Rn is uniquely represented as

n1 ˛ D ˛0 C 2˛1 CC2 ˛n1;˛i 2 Tn .0 Ä i Ä n 1/:

R E E The unit group n is a direct product of a cyclic group h i and where D 1 2 R ˛ t 0 ˛ R f C aja 2 ng. Hence 0 D 6D if is an element of the unit group n .Then ˛ is represented as

t n1 t ˛ D C 2˛1 CC2 ˛n1 D .1 C 2a/;

t t n2 t where a D ˛1 C 2˛2 CC2 ˛n1 . Thus we recognize a is an element of Rn1. The ring automorphism f W Rn ! Rn as ˛f ˛2 2˛2 2n1˛2 D 0 C 1 CC n1 is called a Frobenius automorphism. We deﬁne the relative trace Tn from Rn to An as

f f s1 Tn.˛/ D ˛ C ˛ CC˛ :

We deﬁne the homomorphism nl W Rn ! Rnl by

Xs1 Xs1 i i nl. QiQ / D i iD0 iD0

nl where i ÁQi .mod 2 /, Qi 2 An and i 2 Anl. The commutative relation

nlTn D Tnl nl holds. For more details, we refer the reader to [12].

Lemma 2.1 ([16]). The additive characters of Rn are given by

Tn.ˇ˛/ ˇ.˛/ D 2n

n where ˇ 2 Rn and 2n is a primitive 2 th root of unity. 238 M. Yamada

n In whatP follows, we denote a primitiveP 2 th root of unity by 2n .Welet 1.R / D ˛ R 1.˛/ and 1.pn/ D ˛ p 1.˛/. n 2 n 2 n Lemma 2.2 ([16]). For a nontrivial additive character 1, we have

.R/ 0 .p / 0: 1 n D and 1 n D

3 A Necessary and Sufﬁcient Condition

Let DnC1 Â RnC1: We deﬁne the elements X X D ˛ D 1 . ˛/ nC1 D and nC1 D ˛2DnC1 ˛2DnC1 of the group ring ZRnC1. The subset DnC1 of RnC1 is a difference set with parameters

. 1/ .nC1/s 1 .nC1/s .nC1/s 1 .nC1/s 1 v D 2 nC s; k D 2 2 .2 2 1/; D 2 2 .2 2 1/ if and only if .n C 1/s is even and X D D 1 . /0 ˛ nC1 nC1 D k C ˛2RnC1 where 0 is the zero element of the additive group of RnC1. We call k the order of a difference set. It suffices to show that for every additive character ˇ of RnC1 1 k ; if ˇ ¤ 0; ˇ.D 1D / D nC nC1 k C v D k2; if ˇ D 0 holds. If ˇ D 0,then 0.DnC1/ has to be jDnC1jDk.Forˇ ¤ 0, ˇ.DnC1/ is an element of the integer ring of the cyclotomic field Q.2nC1 /. Since the principal ideal . ˇ.DnC1// is equal to the principal ideal . ˇ.DnC1// and the ideal .2/ is completely ramified in the integer ring of Q.2nC1 /, the ideal . ˇ.DnC1// is equal .nC1/s 1 to the ideal .2 2 /,or

.nC1/s 2 1 ˇ.DnC1/ D 2 u where u is a unit of Q.2nC1 /. Menon–Hadamard Difference Sets Obtained from a Local Field 239

4 Local Fields

Let p be a prime number and denote the p-adic absolute value by jjp.LetQp be the completion of the rational field Q under the p-adic absolute value and Zp be a valuation ring of Qp.LetfQ.x/ be a monic irreducible polynomial of degree s over Zp ps1 n which divides x 1. Assume that fG.x/ Á fQ.x/.mod p / be a basic primitive n polynomial over GR.p ; s/ and fF.x/ Á fQ.x/.mod p/ be a primitive polynomial over GF.ps/. We consider the extension K D Qp./ by adjoining , a root of fQ.x/.The extension K is complete with respect to the unique extended absolute value jjof the p-adic absolute value jjp.Asthep-adic field Qp is a local field, the algebraic extension K is also a local field and the splitting field of fQ.x/. Then we note that the prime element of K is p. s The Galois group of K=Qp is isomorphic to the Galois group of GF.p /=GF.p/, hence a cyclic group of order s.Leth i be the Galois group of K.Wedefinethe relative trace of ˛ from K to Qp as

.˛/ ˛ .˛/ s1.˛/: TK=Qp D C CC

5Ap-Adic Logarithm Function

OK Dfx 2 K WjxjÄ1g is the valuation ring of K and pK Dfx 2 K Wjxj <1g is the maximal ideal of OK .Wedefineap-adic logarithm function. 1 O Definition 5.1. Let B D C p K.Wedefineap-adic logarithm function logp W B ! pOK as

X1 xj log .1 C x/ D .1/jC1 p j jD1 for x 2 pOK. The p-adic logarithm function satisﬁes the following equation (see Proposi- tion 4.5.3. in [4]),

.1 /.1 / .1 / .1 /: logp C x C y D logp C x C logp C y

In what follows, we let K D Q2./. We have the following lemma.

Lemma 5.1. Let OK be a valuation ring of K D Q2./.

1. The 2-adic logarithm function log2 from 1 C 2OK to 2OK is a homomorphism and the kernel of log2 is f1; 1g. 2 2 2. We restrict log2 to 1 C 2 OK.Thenlog2 is an isomorphism from 1 C 2 OK to 2 2 OK . 240 M. Yamada

6 Formal Groups

Let R Pbe a commutative ring withP an identity. We denote the set of formal power 1 n ŒŒ 1 n m ŒŒ ; series nD0 anX by R X and n;mD0 an;mX Y by R X Y . Deﬁnition 6.1. A formal group over R is a formal power series F.X; Y/ satisﬁes the following properties: 1. F.X; Y/ Á X C Y .mod deg 2/, 2. F.X; Y/ D F.Y; X/, 3. F.X; F.Y; Z// D F.F.X; Y/; Z/.

Lemma 6.1. Let K D Q2./. A power series H.X; Y/ D X CY C2XY 2 OK ŒŒX; Y is a formal group over OK . Proof. We easily check the conditions in Deﬁnition 6.1. ut We introduce a homomorphism between two formal groups. Deﬁnition 6.2. A homomorphism h W F ! G between two formal groups is a power series h.X/ 2 RŒŒX with h.0/ D 0 such that

h.F.X; Y// D G.h.X/; h.Y//:

P .1 2 / 1 . 1/jC1 .2x/j We consider log2 C x D jD1 j as a formal power series.

Lemma 6.2. Denote the additive formal group over OK by Ga.X; Y/ D X C Y. A homomorphism h from H.X; Y/ to Ga.X; Y/ is given by 1 h.x/ .1 2x/: D 2 log2 C

. 2 / 1 .1 2. 2 // 1 .1 2 /.1 2 / Proof. From h xCyC xy D 2 log2 C xCyC xy D 2 log2 C x C y , we have 1 1 h.x y 2xy/ .1 2x/ .1 2y/ h.x/ h.y/: C C D 2 log2 C C 2 log2 C D C

7 A New Operation

We see that H.˛; ˇ/ converges in OK for ˛; ˇ 2 OK. Then we deﬁne a new operation of OK by the formal group H.X; Y/ as follows: ˛ ˇ D H.˛; ˇ/: Menon–Hadamard Difference Sets Obtained from a Local Field 241

O O The operation deﬁnes a new abelian group structure on K. We denote it by K . O .2n; / ˛;ˇ O Let n W K ! GR s be the natural projection. For 2 K ,wedeﬁnea new operation by

n.˛/ n.ˇ/ D n.˛ ˇ/ and it is easily verify that this operation is well-deﬁned. Then the Galois ring GR.2n; s/ forms an abelian group with respect to this operation. The additive formal n group Ga.X; Y/ introduces the ordinary additions of OK and GR.2 ; s/.

8 A Family of Menon–Hadamard Difference Sets

We ﬁx an integer m 0 and assume n is odd. Let T be a set of complete O =p pm 2mOC representatives of K K. We deﬁne an additive subgroup of the ideal K D K as follows:

m X.m/ Df2 u j TK=Q2 .u/ Á 0.mod 2/g:

. / pj . / We deﬁne a subset Xm j of K by using X m ,thatis [ [ [ m1 m2 j Xm.j/ D .X.m/ C 2 ˛1 C 2 ˛2 C2 ˛mj/

˛12T ˛22T ˛mj2T for 0 Ä j Ä m 1: Furthermore put

. / 1. . // 2jO: Ym j D h Xm j K

Let m D n 2l 1 and put

. 2 1/ . .0// R : V n l D nl Yn2l1 nl

l lC1 We deﬁne the subset Dp l of p 1 p 1 as nC1 nC nC

2[s1 [ l t Dp l D 2 .1 C 2˛/ nC1 tD0 ˛2V.n2l1/

Theorem 8.1. Assume that n is an odd positive integer. The subset

.n[1/=2 DnC1 D Dp l nC1 lD0 242 M. Yamada

. 1/ .nC1/s 1 .nC1/s is a Menon–Hadamard difference set with v D 2 nC s; k D 2 2 .2 2 1/; .nC1/s 1 .nC1/s 1 and D 2 2 .2 2 1/. We obtain a family of Menon–Hadamard difference sets by ﬁxing s and varying n and l. This family has a nested structure, that is, DnC1 is embedded in the ideal part of DnC3, 2DnC1 Â DnC3.

9 Gauss Sums

R R Let Q be a character of the ordinary multiplicative group nC1 of nC1. We assume the order of Q is a power of 2.Then./Q D 1 and

.Q t.1 C 2˛/ u.1 C 2ˇ// DQ..1 C 2˛/.1 C 2ˇ// DQ.1 C 2.˛ ˇ//

t.1 2˛/ u.1 2ˇ/ R R for C , C 2 nC1. Hence a multiplicative character Q of nC1 R can be regarded as a multiplicative character of the group n .Weextend Q as the character of RnC1 by deﬁning .˛/Q D 0 for any element ˛ 2 pnC1. Denote the 0 trivial character by Q . For a multiplicative character Q and an additive character ˇ of RnC1,wedeﬁnetheGausssumoverRnC1 by X G.;Q ˇ/ D .˛/Q ˇ .˛/:

˛2RnC1

Gauss sums has the following relation. h u Lemma 9.1 ([16]). For ˇ D 2 .1 C 2ˇ0/ 2 RnC1, we have ˇ 1 G.;Q ˇ/ DQ . /G.;Q 2h /: 2h

10 The Determination of Gauss Sums

R We deﬁne a multiplicative character of nl as follows: 1 if ˛0 2 V.n 2l 1/; .˛0/ D 1 if ˛0 … V.n 2l 1/

˛ R R for 0 2 nl and deﬁne the multiplicative character Q of nC1l as follows:

t .˛/Q DQ. .1 C 2˛0// D .˛0/: Menon–Hadamard Difference Sets Obtained from a Local Field 243

We extend Q as the character of RnC1l by deﬁning .˛/Q D 0 for any element ˛ 2 pnC1l. The characteristic function of Dp l is nC1 ( X1 1 1 if ˛ 2 Dp l ; Qj.˛/ D nC1 2 0 if ˛ ¤ Dp l : jD0 nC1

For an additive character ˇ,

1 X1 X 1 X1 j j ˇ.Dp l / D Q .˛/ ˇ.˛/ D G.Q ; ˇ/: nC1 2 2 jD0 ˛2RnC1l jD0

j In order to prove Theorem 8.1, we must determine the values of G.Q ; ˇ/.

Theorem 10.1. Put R D RnC1 and p D pnC1. The following table shows the 0 values of G.;Q ˇ/; G.Q ; ˇ/, where u is a unit of Q.2nC1l /.

0 ˇ G.;Q ˇ/ G.Q ; ˇ/ R pl 00 C1 nC1 pl pl 2 2 su 0 plC1 pnl 00 pnl pnlC1 0 2.nl/s pnlC1 02nl.2s 1/

Proof. Put M D n C 1 l. .1/ ˇ R pl ˇ 2ht.1 2ˇ / 0 < Assume 2 nC1 nC1 and put D C 0 , Ä h l.From j Lemma 9.1, it sufﬁces to determine the values of G.Q ; 2h /. X X m 2 j j NG DjG.Q ; 2h /j D Q ./ 2h ./ Q .ı/ 2h .ı/ R ı R 2 M 2 M X X j j D Q .ı/ 2h .ı/ Q .ı/ 2h .ı/ R ı R 2 M 2 M X X j D Q ./ 2h .. 1/ı/: R ı R 2 M 2 M

P P h 2 TM ..1/ı/ .. 1/ı/ We calculate the inner sum ı R 2h D ı R M . 2 M 2 M 2 2h. 1/ pM1 1 2u ;0 < Assume that … M . We put D C 0 Ä u C h M 1 R , 0 2 Mu. From the commutativity between the trace function and the homomorphism, MuhTM D TMuh Muh mentioned in Sect. 2, 244 M. Yamada X X X 2u . ı/ . ı/ .. 1/ı/ TM 0 TMuh 0 2h D 2Mh D 2Muh ı2R ı2R ı R M M 2 Muh

where ı D Muh.ı/; 0 D Muh.0/. Thus we obtain X 2h .. 1/ı/ D 0 ı R 2 M

2h. 1/ pM1 pM from Lemma 2. Next let 2 M M,wehave

X R X j Mj tr.0ı/ .nl/s 2h .. 1/ı/ D .1/ D2 jGF.2s/j ı R s 2 M ı2GF.2 /

.2s/ .2/ 2h. 1/ pM where tr is the absolute trace from GF to GF .If 2 M,then X .. 1/ı/ R 2.nl/s.2s 1/: 2h Dj MjD ı R 2 M

Hence X X 2h .. 1/ı/ .. 1/ı/ TM 2h D 2M ı R ı R 2 M 2 M 8 0 1 pMh1; < if … M 2.nl/s 1 pMh1 pMh; D : if 2 M M 2.nl/s.2s 1/ 1 pMh: if 2 M

1 2Mh1 pMh1 pMh R Assume that D 0 2 M M with 0 2 hC1.There 2Mh2Q 2nlh1Q 2nlh1O 2n2l1O exists an element 0 D 0 2 K K such nlh1 nlh1 nlh1 n2l1 lh that M.2 Q0/ D 2 0.Since2 Q0 D 2 .2 Q0/ and lh Q nlh1 Q TK=Q2 .2 0/ Á 0.mod 2/, and from Lemma 5.1, 2 0 2 X.n 2l 1/, nlh1 then 2 0 2 V.n 2l 1/. Hence X X j j nlh1 hs s Q ./ D .2 0/ D 2 .2 1/: 1 1 pM h pM h 0 R 2 M M 2 hC1

Mh lh1 Q If 1 2 2 0 with 0 2 Rh, then from TK=Q2 .2 0/ Á 0.mod 2/, X X j j nlh hs Q ./ D .2 0/ D 2 : 1 pMh 02R 2 M h Menon–Hadamard Difference Sets Obtained from a Local Field 245

Thus we obtain

.nl/s hs s .nl/s s hs NG D2 2 .2 1/ C 2 .2 1/2 D 0;

j then it follows G.Q ; 2h / D 0 for j D 0; 1. .2/ ˇ 2lt.1 2ˇ / pl plC1 Assume that D C 0 2 M M . It sufﬁces to determine the value of G.Qj; 2l /. Similarly to the case .1/,wehave 8 1 X < 0 if 1 … pMl ; 2l .. 1/ı/ M TM 2.nl/s 1 pMl1 pMl; 2M D : if 2 M M ı R 2.nl/s.2s 1/ 1 pMl: 2 M if 2 M

1 2Ml1 pMl1 pMl If D 0 2 M M ,then X X j j n2l1 Q ./ D .2 0/ 1 1 pM l pM l 0 R 2 M M 2 lC1 X X j n2l1 j n2l1 D .2 0/ .2 0/:

02RlC1 02p lC1

n2l1 n2l1 n2l1 n2l1 Let 2 Q0 2 2 OK such that M .2 Q0/ D 2 0.IfQ0 2 pK, n2l1 n2l1 then 2 Q0 2 X.n 2l 1/ and 2 0 2 V.n 2l 1/. Thus X j n2l1 ls .2 0/ D 2 :

02p lC1

If Q0 runs through OK,then0 D M.Q0/ runs through RlC1. Therefore X .lC1/s j n2l1 2 if j D 0, .2 0/ D 0 if j D 1. 02RlC1

Thus we have X 2ls.2s 1/ if j D 0, Qj./ D 2ls if j D 1. 1 p Ml1 p Ml 2 M M

1 2Ml pMl Next we assume D 0 2 M . Similarly to the above, X Qj./ D 2ls: 1 p Ml 2 M 246 M. Yamada

Consequently, 2.nl/s 2ls.2s 1/ C 2.nl/s.2s 1/2ls D 0 if j D 0; N D G 2.nl/s.2ls/ C 2.nl/s.2s 1/2ls D 2.nC1/s if j D 1:

Hence

C1 0 n s G.Q ; 2s / D 0; G.;Q 2s / D 2 2 u

C1 n s 0 0 where u is a unit of Q.M/. It follows G.;Q ˇ/ D 2 2 u where u is a unit of Q.M/. .3/ ˇ ph 1 < .2/ Assume 2 nC1, l C Ä h n l. Similarly to the case , 8 1 X < 0 if 1 … pMh ; 2h .. 1/ı/ M TM 2.nl/s 1 pMh1 pMh; 2M D : if 2 M M ı p 2.nl/s.2s 1/ 1 pMh: 2 M if 2 M

1 pMh1 pMh Let 2 M M . X X j j nlh1 Q ./ D .2 0/ 1 1 p M h p M h 02R 2 M M hC1 X X j nlh1 j nlh1 D .2 0/ .2 0/:

02RhC1 02p hC1

nlh1 n2l1 Notice that 2 OK 2 OK. If the element Q0 runs through OK , then 0 D M.Q0/ runs through RhC1. Therefore X nlh1 .2 0/ D 0

02RhC1

and also X nlh1 .2 0/ D 0:

02p hC1

Then X X ./Q D 0; and Q0./ D 2hs.2s 1/: 1 p Mh1 p Mh 1 p Mh1 p Mh 2 M M 2 M M Menon–Hadamard Difference Sets Obtained from a Local Field 247

1 pMh If 2 M ,then X 2hs if j D 0; Qj./ D 0 if j D 1: 1 p Mh 2 M

j j It follows NG D 0; that is, G.Q ; 2h / D G.Q ; ˇ/ D 0: We assume h D nl t M 1 D n l and put ˇ D 2 .1 C 2ˇ0/.Wehave X 2nl .t.1 2ˇ // .j; / j./ TM C 0 G Q ˇ D Q 2M R 2 M X 2Xs1 j Tr.u.1C2ˇ0/.1C20// D .0/ .1/

02RM1 uD0 2.nl/s if j D 0; D 0 if j D 1:

If h M,then X nl s j j 2 .2 1/ if j D 0; G.Q ; ˇ/ D Q ./ D 0 if j D 1: R 2 M It completes the proof of the theorem. ut

11 The Proof of the Main Theorem

. C1/ n s 1 As mentioned in Sect. 3,weverify ˇ.D/ D 2 2 u if ˇ 6D 0 and 0.D/ D . C1/ . C1/ n s 1 n s 2 2 .2 2 1/ if ˇ D 0,whereu is a unit of Q.2nC1 /.SinceX.m/ is a 2mOC 2 . /; 0 1 2lO subgroup of K with index , Y j Ä j Ä m is a subgroup of K with index 2. It leads jV.n 2l 1/jD2.nl/s1. Thus

n1 2 X . C1/ . C1/ .nl/s1 s n s 1 n s 0.D/ D 2 .2 1/ D 2 2 .2 2 1/: lD0

We denote the corresponding elements of DR and Dp l by DR and Dp l . nC1 nC1 nC1 nC1 n1 We display the values ˇ.DR /, ˇ.Dp l /; 1 Ä l Ä 2 ,inTable1 from nC1 nC1 Theorem 10.1, which is a similar table as in the paper [16]. For convenience, we put R R pl pl . / D nC1 and D nC1 and denote the units of Q 2nC1 by the same character u even though these are different. 248 M. Yamada

n1 Table 1 The values of ˇ.DR /, ˇ.Dp l /; 1 Ä l Ä nC1 nC1 2

ˇ ˇ.DR / ˇ .Dp/ ˇ .Dp l / ˇ .D n3 / ˇ .D n1 / p 2 p 2 nC1 1 R ˙2 2 s u 0000 2 nC1 1 p p 0 ˙2 2 s u 000 : : : : : : : : : : : : : : : : : : C1 nC1 1 p l p l 00˙2 2 s u 00 : : : : : : : : : : : : : : : : : : n3 n1 nC1 1 p 2 p 2 00 0˙2 2 s u 0 n1 nC1 nC1 1 p 2 p 2 00 0 022 s u nC1 nC3 nC1 1 p 2 p 2 00 0 02 2 s nC3 nC5 nC3 1 nC1 1 p 2 p 2 00 02 2 s 2 2 s .2s 1/ : : : : : : : : : : : : : : : : : : C1 . / 1 nC3 1 nC1 1 p n l p n l 002 n l s 2 2 s .2s 1/ 2 2 s .2s 1/ : : : : : : : : : : : : : : : : : : 1 . 1/ 1 . / 1 nC3 1 nC1 1 p n p n 0 2 n s 2 n l s .2s 1/ 2 2 s .2s 1/ 2 2 s .2s 1/ 1 . 1/ 1 . / 1 nC3 1 nC1 1 p n f0g2ns 2 n s .2s 1/ 2 n l s .2s 1/ 2 2 s .2s 1/ 2 2 s .2s 1/

C1 n s1 It is easily verified that ˇ.D/ D 2 2 u for ˇ 6D 0 where u is a unit of a cyclotomic field Q.2nC1 / from the above table. pl plC1 R 2lt.1 2 / 2lC1 The element of nC3 nC1 of nC3 is represented as C a and a 2 plC1 plC1 2lC3R R nC3.Since nC3 D nC3, we can regard a as an element of nC1. The subset defining Dpl is V.n C 2 2l 1/ D V.n C 1 l/ and the subset defining Dpl1 is nC3 nC1 . 2. 1/ 1/ . 1 / 2 ;1 nC1 also V n l D V nC l . Thus we have Dpl1 Â Dpl Ä l Ä 2 . nC1 nC3 Hence 2DnC1 DnC3. Notice that resultant difference sets and the constructions are different from them in the paper [16]. There still remain some problems to know, for example, whether the difference set over a 2-group, especially McFarland difference set, is equivalent to our difference set or not. We think this construction method is applicable to the other topics of combinatorics, for example, design theory, coding theory, etc.

Acknowledgements The author would like to thank the referees for the careful reading and the valuable suggestions. This work was supported by JSPS KAKENHI Grant Number 24540013.

References

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3. Davis, J., Jedwab, J.: A survey of Hadamard difference sets. In: Arasu, K.T., et al. (eds.) Groups, Difference Sets and the Monster, pp. 145–156. de Gruyter, Berlin-New York (1996) 4. Gouvêa, F.Q.: p-adic Numbers. An Introduction. Springer, Berlin-Heidelberg (1997) t 5. Hou, X.-D., Leung, K.H., Xiang, Q.: New partial difference sets in Zp2 and a related problem about Galois rings. Finite Fields Appl. 7, 165–188 (2001) 6. Jungnickel, D.: Difference sets. In: Dinitz, J.H., Stinson, D.R. (eds.) Contemporary Design Theory, pp. 241–324. Wiley, New York (1992) 7. Jungnickel, D., Pott, A., Smith, K.W.: Difference sets. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, pp. 419–435. Chapman & Hall/CRC, New York (2007) 8. Koblitz, N.: p-Adic Analysis. A Short Course on Recent Work. Cambridge University Press, New York (1980) 9. Kraemer, R.G.: Proof of a conjecture on Hadamard 2-groups. J. Comb. Theory Ser. A 63, 1–10 (1993) 10. Lagorce, N.: A convolutional-like approach to p-adic codes. Discrete Appl. Math. 1114, 139–155 (2001) 11. Ma, L.K.: A survey of partial difference sets. Des. Codes Crypt. 4, 221–261 (1994) 12. McDonald, B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974) 13. Neukirch, J.: Class Field Theory. Springer, Berlin-Heidelberg (1986) 14. Polhill, J.B.: Constructions of nested partial difference sets with Galois rings. Des. Codes Crypt. 25, 299–309 (2002) 15. Ray-Chaudhuri, D.K., Xiang, Q.: Constructions of partial difference sets and relative difference sets using Galois rings. Des. Codes Crypt. 8, 215–227 (1996) 16. Yamada, M. : Difference sets over the Galois rings with odd extension degree and characteristic an even power of 2. Des. Codes Crypt. 67, 37–57 (2013) BIRS Workshop 14w2199 July 11–13, 2014 Problem Solving Session

R. Craigen (Problems Editor)

Abstract A companion event to the Lethbridge July 8–11 meeting Workshop on Algebraic Design Theory and Hadamard Matrices, the 2014 BIRS July 11–13 meeting Algebraic design theory with Hadamard matrices: applications, current trends and future directions, culminated in a lively problem session considering problems raised in both meetings. This session was unusually productive. By general consensus the problems posed comprise a provocative collection with great potential to incite new and interesting work. It captured well the elusive spirit generally sought in conference problem sessions: a mixture of on-the-spot group collaboration, instant insights and stubborn conundrums that could shift the focus of the general research community in helpful ways, reﬂecting the current state of the discipline and a snapshot of the kind of questions people are thinking about “on the ground” today. Perhaps most who attended went home having squirrelled away a problem or two from this session, to bring out on a rainy (or snowy) day. Several of the problems, with background information, were submitted in the form of brief notes for publication, and these are compiled here, edited somewhat for space. They represent a wide spectrum of both subject matter and types of queries. There are good problems for students to work on—being accessible and likely to hive off special cases for tractable small projects; and there are difﬁcult research problems that will probably elude solutions in the foreseeable future.

Keywords Hadamard matrices • Generalized Hadamard matrices • Modular Hadamard matrices • Algebraic design theory • Central relative difference sets

R. Craigen () Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada MB R3T 2N2 e-mail: [email protected]

© Springer International Publishing Switzerland 2015 251 C.J. Colbourn (ed.), Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics & Statistics 133, DOI 10.1007/978-3-319-17729-8_21 252 R. Craigen

1 Case Study for Modular H.668/

Submitted by R. Craigen, University of Manitoba. The suggestion arose after F. Szöllosi’s˝ talk that there would be value in survey- ing the existence of mod m Hadamard matrices of order 668 for values of m Ä 668, and that individual cases would provide helpful exercises for undergraduate or graduate students still finding their feet in this field while potentially unearthing new insight about the relation between modular and “ordinary” Hadamard matrices and making ostensible progress toward settling the smallest unresolved case of existence of Hadamard matrices. For clarity: define a mod m Hadamard matrix of order n tobeamatrixH 2 f˙1gnn such that HH> Á nI .mod m/, where the congruence applies entry-wise. Results presented in Szöllosi’s˝ talk imply existence for m D 2;3;5 and 6. Marrero and Butson [14] is an early introduction to the subject. Eliahou and Kervaire [8] establishes existence for m D 64 and thus also 32, 16, 8 and 4. It also establishes the case m D 12, and perhaps others.

2 Twin Strongly Regular Graphs: Some Questions

Submitted by Paul Leopardi, Australian National University. A simple graph of order v is strongly regular [4] with parameters .v; k;;/if • each vertex has degree k; • each adjacent pair of vertices has common neighbours, and • each nonadjacent pair of vertices has common neighbours. Question 2.1. For which parameters .v; k;;/does there exist a regular graph G of order v and degree 2k that can be given a two-edge colouring (say red and blue) such that each of the red and blue subgraphs are strongly regular with parameters .v; k;;/and such that there exists an automorphism of G that swaps the two edge colours?

Example 2.1. The two-edge-coloured graphs m; m 1; deﬁned in [13], form a sequence where each of the red and blue subgraphs of m are strongly regular with parameters

. ; k;D / D .4m;22m1 2m1;22m2 2m1/:

For m D 1; 2; 3 it is relatively easy (e.g. using iGraph) to construct an automorphism of m that swaps the two colours. For m >3the problem is open.

Figure 1 shows 2; with each of the red and blue subgraphs being a .16; 6; 2; 2/ strongly regular graph. (Please ignore the vertex colouring.) BIRS Workshop 14w2199 July 11–13, 2014 Problem Solving Session 253

Fig. 1 2

Question 2.2. As a special case of Question 2.1, restrict . ; k;;/ to D 4m; k D 22m1 2m1;D D 22m2 2m1: In particular, for which m is there an isomorphism of m that swaps red and blue edges?

3 Irredundant Orthogonal Arrays

Submitted by Dardo Goyeneche and Karol Zyczkowski˙ , Jagiellonian University. Consider a standard orthogonal array with r runs, N factors, d levels, strength k and index D r=dk, denoted OA(r; N; d; k). Deﬁnition 3.1. An OA(r; N; d; k) is called irredundant, written IrOA if, when any k columns are removed from the array, all rows of the resulting r .N k/ array are distinct. Problem 3.1. List and characterize irredundant orthogonal arrays of low orders. Notice that any OA of index D 1 is by construction irredundant. Thus one has to deal with the cases 2 and can start the problem analysing arrays with index two. Problem 3.2. Asshownin[9] each irredundant array OA(r; N; d; k) corresponds to H ˝N .2;3;2;1/ a k-uniform quantum state of N qudits, j i2 d . For instance, OA corresponds to the 3-qubit GHZ state,

jGHZ3iDj000iCj111i; 254 R. Craigen while OA.4;3;2;2/represents the state

j˚3iDj000iCj011iCj101iCj110i:

It can be shown that these states are locally equivalent, written jGHZ3i loc j˚3i, i.e. there exists a product unitary matrix Uloc (in this case the Hadamard H8 D H2 ˝ H2 ˝ H2) such that jGHZ3iDUlocj˚3i. Therefore a natural question arises: What pairs of IrOA lead to locally equivalent quantum states? Note that the number r of terms in each state (i.e. the number of runs in the array) needn’t be preserved by the local equivalence relation. Moreover, the strength k of the two arrays needn’t be equal, as shown in the case above.

4 Almost Hadamard Matrices and Flat Orthogonal Sequences

Submitted by Mate Matolcsi and Karol Zyczkowski˙ Consider the following analytic variants of the Hadamard conjecture. T An almost Hadamard matrixP A of order N is an orthogonal matrix, AA D N1,such that its 1-norm kAk1 D jai;jj is a local maximum of the 1-norm on the orthogonal group O.N/. A is called optimal if kAk1 is a global maximum of the 1-norm on O.N/ (see [1–3]). Clearly, a Hadamard matrix H is an optimal almost Hadamard matrix.

If Ni is any strictly increasing sequence of positive integers, a sequence UNi of T 1 orthogonal matrices, UNi UN D Ni , is called a ﬂat orthogonal sequence if each 1i .1/ entry of UNi is of the form ˙ Co (see [11]). Any sequence of Hadamard matrices

HNi (where Ni !1is divisible by 4) is a ﬂat orthogonal sequence, of course.

Problem 4.1. Find a ﬂat orthogonal sequence UN where N ranges through all dimensions. We conjecture that such sequence exists.

Problem 4.2. Decide whether a ﬂat sequence of circulant orthogonal matrices CN exists (again N ranges through all dimensions).

Problem 4.3. It is trivial that optimal almost Hadamard matrices AN exist for all N [by compactness of O.N/]. Decide whether a sequence AN of optimal almost Hadamard matrices is automatically a ﬂat orthogonal sequence.

5 Alltop Problems

Submitted by Asha Rao, RMIT University. Let f be a mapping from a ﬁnite Abelian group G to another ﬁnite Abelian group H. For each element a 2 H,thedifference function of f with respect to a is

f ;a.x/ D f .x C a/ f .x/ BIRS Workshop 14w2199 July 11–13, 2014 Problem Solving Session 255

Often the orders of the two groups G and H are the same, but this is not necessary. Given finite groups G and H of the same order, a function f W G ! H is called a planar function if the difference function of f is a bijection for every a 2 G.An Alltop Function f is a mapping from a finite group G to another finite group H,such that the difference function f ;a.x/ is a planar function for every a 2 G. For additional background and definitions, see [10]. The following are some known results: • Alltop functions cannot exist on fields of characteristic 3. • For the class of planar functions of the form ˘.x/ D xpkC1, with p aprime, f .x/ D x3 is the only Alltop function. •Letp 5 be an odd prime and r an integer such that 3 does not divide pr C 1. prC2 Then A.x/ D x is an Alltop polynomial on Fp2r . The difference functions of A.x/ are EA-equivalent to x2,butA.x/ is EA-inequivalent to x3. Some open problems follow. Problem 5.1. Are there other families of Alltop functions EA-inequivalent to x3? Problem 5.2. Is the A.x/ given above CCZ inequivalent to x3? Problem 5.3. Do non-equivalent Alltop functions generate non-equivalent MUBs? Problem 5.4. Analyse the properties of this new family of Alltop functions with regard to correlation measure and transmission methods.

6 Fewest Differences Between Two Hadamard Matrices

Submitted by: Padraig Ó Catháin and Ian M. Wanless, Monash University. In [16] the authors showed that any pair of distinct real Hadamard matrices must differ in at least n places. We also showed a similar result for complex Hadamard matrices that differ on a submatrix. Moreover, any rank one submatrix with n entries can be switched to produce a new Hadamard matrix. Question 6.1. Are there even integers a; b for which there does not exist a Hadamard matrix of order ab containing an a b rank one submatrix? On the basis of our results in [16] we are inclined to think that the answer to the following question is negative: Question 6.2. Do there exist two distinct n n complex Hadamard matrices that differ in fewer than n entries? A set of entries of a Hadamard matrix that can be changed to produce a new Hadamard matrix is known as a trade. It would be nice to know how “universal” the rectangular trades we studied are. 256 R. Craigen

Question 6.3. Is every trade in a (real) Hadamard matrix a Z2-linear combination of rectangular trades? Our work was motivated in part by problems in the construction of compressed sensing matrices. Optimal complex Hadamard matrices for this application have the property that linear combinations of t rows vanish in at most t components. Question 6.4. Other than Fourier matrices, are there families of Hadamard matrices with the property that no linear combination of t rows contains more than t zeros? If such matrices are rare, describe families in which no linear combination of t rows contain more than f .t/ zeros for some slowly growing function f . Lastly, there is the issue of generalising to weighing matrices. Question 6.5. Can two different weighing matrices of the same order and weight differ in fewer positions than their weight?

7 A Question About the Largest Power of 2 Dividing per H.n/

Submitted by Ian Wanless, Monash University. The permanent of an n n matrix A D Œai;j is deﬁned by

X Yn per.A/ D ai;.i/;

2Sn iD1

k with Sn the symmetric group on f1;2;:::;ng.Letf .n/ D maxfk 2 Z W 2 divides nŠg.In[17] it was reported that every Hadamard matrix H of order n <32has the property that per.H/ is divisible by 2f .n/. Except in the trivial case where n D 2,it was also found that per.H/ is not divisible by 2f .n/C1. The same property holds for all Hadamard matrices of order 32 and type 2 or 3. By a result of Kräuter and Seifter [12] we know that 2nblog2.nC1/c divides per.H/. Conjecture 7.1. A Hadamard matrix H of order n 4 has per.H/ divisible by 2f .n/ but not by 2f .n/C1. If true, this would solve open problem #5 from Minc’s catalogue [15]ofopen problems on permanents (see also [6]).

8 Central Relative Difference Sets via Pairs of Binary Sequences

Submitted by Ronan Egan, National University of Ireland. BIRS Workshop 14w2199 July 11–13, 2014 Problem Solving Session 257

4t1 4t1 Denote the dicyclic group of order 8t by Q8t Df1; x;:::;x ; y; xy;:::;x yg. Ito has conjectured that there exists a .4t;2;4t;2t/-CRDS (central relative difference set) in Q8t for all positive t. A theorem by De Launey proves that a cocyclic Hadamard matrix exists if and only if there is a .4t;2;4t;2t/-CRDS, and the dicyclic group has been a rich source of these difference sets. We deﬁne the periodic autocorrelation of a f˙1g-sequence a of length n, with shift k to be

Xn Pa.k/ D aiaiCk: iD1 where subscripts are taken modulo n.

Lemma 8.1. The set of all CRDS in Q8t is in one to one correspondence with the set of all pairs of sequences a; b with entries in f˙1g of period 4t satisfying the following two properties:

1. ai DaiC2t and bi DbiC2t for all 1 Ä i Ä 2t. 2. Pa.k/ C Pb.k/ D 0 for all 1 Ä k Ä 2t 1. We call pairs a and b satisfying the properties of Lemma 8.1 suitable pairs. Table 1 is an enumeration n.t/ of the existing suitable pairs a; b for 1 Ä t Ä 8. Question 8.1. For what values of t can we guarantee existence of suitable pairs? (Aside from those arising from known constructions of central relative difference sets in Q8t,Ito,Schmidt,etc.)

Table 1 Suitable pairs t 1 2 3 4 5 6 7 8 n.t/ 16 128 576 4,096 11,200 59,904 90,944 557,056

Question 8.2. Can we create new longer suitable pairs via some construction building on existing shorter suitable pairs? Question 8.3. Is there some sensible methodical way to generate suitable pairs? Or to generate pairs that are suitable with high probability? Can we say anything interesting at all? Denote the aperiodic autocorrelation of a at shift k by

Xnk Aa.k/ D aiaiCk: iD1

The subsequences x D a1 :::a2t and y D b1 :::b2t of a suitable pair a; b of length 4t satisfy:

Ax.k/ C Ay.k/ D Ax.2t k/ C Ay.2t k/: 258 R. Craigen

Thus in searching for pairs a; b of length 4t it sufﬁces to seek sequences x; y of length 2t with this property, whence a D .x; x/; b D .y; y/ are a suitable pair. Golay pairs of length 2t are an example of such sequences, but they are more restrictive and there are others.

9 Linking Systems: Piecing Together Difference Sets for Symmetric Designs

Submitted by William J. Martin, Worcester Polytechnic Institute. Problem 9.1. Find new examples of linking systems of difference sets in finite abelian groups. We use the group ring CŒG for our definitions. Recall that D Â G is identified with the formal sum of its elements in this ring. Given this, it is not hard to verify the classical characterization of a .v; k;/difference set in G as a subset D satisfying

.1/ DD D .k /1G C G where D.1/ Dfd1jd 2 Dg. Definition 9.1. A difference set D G is called reversible if D.1/ D D. ˚Definition 9.2. Let G be« a finite group of order v and let ` 1. A collection D ; j 0 Ä i; j Ä `; i ¤ j of .v; k;/difference sets in G is a .v; k;I `C1/-linking i j .1/ system if there exist ; 2 Z such that Dj;i D Di;j for all i ¤ j;andforall distinct h; i; j 2f0;:::;`g, Dh;iDi;j D Dh;j C .G Dh;j/. The first infinite family of linking sets was discovered by Cameron and Seidel in 1973. Theorem 9.1 (Cameron and Seidel [5]). There is a .22t 32t1;22t 2t1;2tC1 1/-linking system in the elementary abelian group of order 22tC2 for all t 2. In 2014, Davis et al. [7] used Galois rings to construct linking sets. Issues: • the designs have the same parameters as above; • the number of designs in our construction is smaller than the number obtained by Cameron and Seidel; • we cannot prove that ours are not isomorphic to the ones above except in the first two cases. We need to find more examples of linking systems. It is known (essentially due to Cameron) that, in order to have more than one difference set in the collection, the order n D k of the design must be a square. So the first cases to study are .v; k;/D .36; 15; 6/, .v; k;/D .45; 12; 3/ and .v; k;/D .85; 21; 5/. BIRS Workshop 14w2199 July 11–13, 2014 Problem Solving Session 259

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