DOCSLIB.ORG

Algebraic Design Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Mathematical Surveys and Monographs Volume 175 Algebraic Design Theory Warwick de Launey Dane Flannery American Mathematical Society http://dx.doi.org/10.1090/surv/175 Mathematical Surveys and Monographs Volume 175 Algebraic Design Theory Warwick de Launey Dane Flannery American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Jordan S. Ellenberg Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 05-02, 05Bxx, 05E18, 16B99, 20Dxx; Secondary 05-04, 15A24, 16S99, 20B20, 20J06. For additional information and updates on this book, visit www.ams.org/bookpages/surv-175 Library of Congress Cataloging-in-Publication Data De Launey, Warwick, 1958– Algebraic design theory / Warwick De Launey, Dane Flannery. p. cm. — (Mathematical surveys and monographs ; v. 175) Includes bibliographical references and index. ISBN 978-0-8218-4496-0 (alk. paper) 1. Combinatorial designs and configurations. I. Flannery, D. L. (Dane Laurence), 1965– II. Title QA166.25.D43 2011 511.6–dc23 2011014837 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 To Scott Godfrey MD, Richard Lam MD, and Mark Scholz MD — Warwick de Launey To my parents, Lois and Ivan — Dane Flannery Contents Preface ix Chapter 1. Overview 1 1.1. What is a combinatorial design? 1 1.2. What is Algebraic Design Theory? 1 1.3. What is in this book? 2 Chapter 2. Many Kinds of Pairwise Combinatorial Designs 7 2.1. Orthogonality sets 7 2.2. Symmetric balanced incomplete block designs 9 2.3. Hadamard matrices 11 2.4. Weighing matrices 12 2.5. Balanced weighing matrices 13 2.6. Orthogonal designs 14 2.7. Complex Hadamard matrices 16 2.8. Complex generalized Hadamard matrices 18 2.9. Complex generalized weighing matrices 19 2.10. Generalized Hadamard matrices over groups 19 2.11. Balanced generalized weighing matrices 22 2.12. Generalized weighing matrices 23 2.13. Summary 24 Chapter 3. A Primer for Algebraic Design Theory 27 3.1. Groups 27 3.2. Monoids 34 3.3. Group actions 35 3.4. Rings 39 3.5. Matrices 41 3.6. Linear and related groups 42 3.7. Representations 44 Chapter 4. Orthogonality 49 4.1. How many rows can be pairwise Λ-orthogonal? 49 4.2. Non-trivial orthogonality sets 50 4.3. A big picture 51 4.4. Equivalence 54 4.5. Matrices, arrays, and designs 60 Chapter 5. Modeling Λ-Equivalence 63 5.1. A first look at the automorphism group 63 5.2. Ambient rings with a model for Λ-equivalence 65 v vi CONTENTS 5.3. Ambient rings for the familiar orthogonality sets 68 Chapter 6. The Grammian 71 6.1. Orthogonality as a Grammian property 71 6.2. Non-degeneracy 72 6.3. Gram completions and composition of orthogonality sets 73 6.4. The Gram Property and Λ-equivalence 74 Chapter 7. Transposability 77 7.1. The main problems 77 7.2. A functional approach to self-duality 78 7.3. Conjugate equivalence operations 80 7.4. A matrix algebra approach to transposability and self-duality 80 7.5. A different kind of transposable orthogonality set 82 Chapter 8. New Designs from Old 85 8.1. Composition 85 8.2. Transference 93 Chapter 9. Automorphism Groups 99 9.1. Automorphism groups of pairwise combinatorial designs 99 9.2. A class of generalized Hadamard matrices 100 9.3. A bound on the size of the automorphism group 103 9.4. Permutation automorphism groups 105 9.5. Automorphism groups of orthogonal designs 106 9.6. Expanded designs 108 9.7. Computing automorphism groups 112 9.8. The associated design 114 9.9. Associated designs and group divisible designs 116 9.10. An isomorphism for weighing matrices 117 Chapter 10. Group Development and Regular Actions on Arrays 119 10.1. Matrix preliminaries 119 10.2. Group-developed arrays 119 10.3. Regular embeddings 121 10.4. Difference sets and relative difference sets 124 10.5. Group ring equations and associates 127 10.6. Finding all associates of an array 129 10.7. An algorithm for solving Problems 10.2.3 and 10.2.4 131 10.8. Composition via associates 132 Chapter 11. Origins of Cocyclic Development 135 11.1. First derivation 135 11.2. Second derivation 140 11.3. Cocycles for cyclic groups 142 Chapter 12. Group Extensions and Cocycles 145 12.1. Central extensions 145 12.2. Cocycles for product groups 150 12.3. Polycyclic presentations 151 12.4. Cocycles from collection in polycyclic groups 153 CONTENTS vii 12.5. Monomial representations and cocycles 157 Chapter 13. Cocyclic Pairwise Combinatorial Designs 161 13.1. The main definitions 161 13.2. Ambient rings with a central group 162 13.3. Some big problems 164 13.4. Central extensions of a design 164 13.5. Approaches to cocyclic designs 165 Chapter 14. Centrally Regular Actions 167 14.1. Cocyclic forms 167 14.2. A lesser expanded design 167 14.3. A pair of lifting homomorphisms 168 14.4. The lift 169 14.5. Translation 170 14.6. Centrally regular embeddings 171 14.7. Finding cocyclic forms 173 14.8. All the cocycles of a design 176 Chapter 15. Cocyclic Associates 177 15.1. Definition of cocyclic associates 177 15.2. The group ring equation for cocyclic associates 178 15.3. The familiar designs 180 15.4. Cocyclic designs and relative difference sets 181 15.5. Normal p-complements 182 15.6. Existence conditions for cocyclic Hadamard matrices 183 15.7. Cyclotomic rings and circulant complex Hadamard matrices 185 15.8. Composition of cocyclic associates 190 Chapter 16. Special Classes of Cocyclic Designs 195 16.1. Cocyclic Hadamard matrices 195 16.2. Cocyclic weighing matrices 197 16.3. Cocyclic orthogonal designs 198 16.4. A cocyclic substitution scheme 200 16.5. Cocyclic complex Hadamard matrices 201 Chapter 17. The Paley Matrices 203 17.1. Actions of 2-dimensional linear and semilinear groups 203 17.2. The Paley matrices and their automorphism groups 205 17.3. The regular actions 209 Chapter 18. A Large Family of Cocyclic Hadamard Matrices 215 18.1. On the orders covered 215 18.2. A construction for prime powers congruent to 3 (mod 4) 216 18.3. A construction for prime powers congruent to 1 (mod 4) 218 18.4. Plug-in matrices 220 18.5. Proof of the main theorem and a generalization 221 Chapter 19. Substitution Schemes for Cocyclic Hadamard Matrices 223 19.1. General substitution schemes 224 19.2. Number-theoretic constraints 226 viii CONTENTS 19.3. Further results for group-developed plug-in matrices 227 19.4. Inverting action 228 19.5. Trivial action 230 19.6. Complementary pairs and the Cocyclic Hadamard Conjecture 232 19.7. Existence of group-developed complementary pairs 233 Chapter 20. Calculating Cocyclic Development Rules 239 20.1. Introduction to development tables 239 20.2. Development tables for abelian groups 240 20.3. Development tables revisited 241 20.4. Group cohomology 242 20.5. Constructing a free table 243 20.6. Group homology 244 20.7. Presentations and the Schur multiplier 246 20.8. Constructing a torsion table 249 20.9. Listing the elements of the second cohomology group 253 20.10. Another look at the Cocyclic Hadamard Conjecture 255 Chapter 21. Cocyclic Hadamard Matrices Indexed by Elementary Abelian Groups 257 21.1. Motivation: indexing groups for the Sylvester matrices 257 21.2. The extension problem 258 21.3. Pure Hadamard collection cocycles 261 21.4. Bilinearity and Hadamard cocycles 262 21.5. Solution of the Hadamard cocycle problem 263 Chapter 22. Cocyclic Concordant Systems of Orthogonal Designs 267 22.1. Existence and uniqueness of cocyclic systems of OD(n;1k) 267 22.2. A reduction 268 22.3. Solution of the reduced problem 269 22.4. Proof of Theorem 22.1.1 270 22.5. Removing the zeros 271 22.6. Examples 272 Chapter 23. Asymptotic Existence of Cocyclic Hadamard Matrices 279 23.1. Complex sequences with zero aperiodic autocorrelation 279 23.2. Sets of Hermitian and skew-Hermitian circulant matrices 281 23.3. Sets of cocyclic signed permutation matrices 282 23.4. Existence of cocyclic complex Hadamard matrices 283 23.5. Concluding remarks 284 Bibliography 287 Index 295 Preface Over the past several decades, algebra has become increasingly important in combinatorial design theory. The flow of ideas has for the most part been from algebra to design theory. Moreover, despite our successes, fundamental algebraic questions in design theory remain open. It seems that new or more sophisticated ideas and techniques will be needed to make progress on these questions. In the meantime, design theory is a fertile source of problems that are ideal for spurring the development of algorithms in the active field of computational algebra.
Recommended publications
  • Combinatorial Design University of Southern California Non-Parametric Computational Design Strategies

    Combinatorial Design University of Southern California Non-Parametric Computational Design Strategies

    Jose Sanchez Combinatorial design University of Southern California Non-parametric computational design strategies 1 ABSTRACT This paper outlines a framework and conceptualization of combinatorial design. Combinatorial 1 Design – Year 3 – Pattern of one design is a term coined to describe non-parametric design strategies that focus on the permutation, unit in two scales. Developed by combinatorial design within a game combination and patterning of discrete units. These design strategies differ substantially from para- engine. metric design strategies as they do not operate under continuous numerical evaluations, intervals or ratios but rather finite discrete sets. The conceptualization of this term and the differences with other design strategies are portrayed by the work done in the last 3 years of research at University of Southern California under the Polyomino agenda. The work, conducted together with students, has studied the use of discrete sets and combinatorial strategies within virtual reality environments to allow for an enhanced decision making process, one in which human intuition is coupled to algo- rithmic intelligence. The work of the research unit has been sponsored and tested by the company Stratays for ongoing research on crowd-sourced design. 44 INTRODUCTION—OUTSIDE THE PARAMETRIC UMBRELLA To start, it is important that we understand that the use of the terms parametric and combinatorial that will be used in this paper will come from an architectural and design background, as the association of these terms in mathematics and statistics might have a different connotation. There are certainly lessons and a direct relation between the term ‘combinatorial’ as used in this paper and the field of combinatorics and permutations in mathematics.
  • The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: Gasarch@Cs.Umd.Edu

    The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: [email protected]

    The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 email: [email protected] In this column we review the following books. 1. Combinatorial Designs: Constructions and Analysis by Douglas R. Stinson. Review by Gregory Taylor. A combinatorial design is a set of sets of (say) {1, . , n} that have various properties, such as that no two of them have a large intersection. For what parameters do such designs exist? This is an interesting question that touches on many branches of math for both origin and application. 2. Combinatorics of Permutations by Mikl´osB´ona. Review by Gregory Taylor. Usually permutations are viewed as a tool in combinatorics. In this book they are considered as objects worthy of study in and of themselves. 3. Enumerative Combinatorics by Charalambos A. Charalambides. Review by Sergey Ki- taev, 2008. Enumerative combinatorics is a branch of combinatorics concerned with counting objects satisfying certain criteria. This is a far reaching and deep question. 4. Geometric Algebra for Computer Science by L. Dorst, D. Fontijne, and S. Mann. Review by B. Fasy and D. Millman. How can we view Geometry in terms that a computer can understand and deal with? This book helps answer that question. 5. Privacy on the Line: The Politics of Wiretapping and Encryption by Whitfield Diffie and Susan Landau. Review by Richard Jankowski. What is the status of your privacy given current technology and law? Read this book and find out! Books I want Reviewed If you want a FREE copy of one of these books in exchange for a review, then email me at gasarchcs.umd.edu Reviews need to be in LaTeX, LaTeX2e, or Plaintext.
  • Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference

    Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference

    Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference Sunday 30 May to Friday 4 June 2010 Index of Talks and Abstracts Main talks 1. Simeon Ball, On subsets of a finite vector space in which every subset of basis size is a basis 2. Simon Blackburn, Honeycomb arrays 3. G`abor Korchm`aros, Curves over finite fields, an approach from finite geometry 4. Cheryl Praeger, Basic pregeometries 5. Bernhard Schmidt, Finiteness of circulant weighing matrices of fixed weight 6. Douglas Stinson, Multicollision attacks on iterated hash functions Short talks 1. Mari´en Abreu, Adjacency matrices of polarity graphs and of other C4–free graphs of large size 2. Marco Buratti, Combinatorial designs via factorization of a group into subsets 3. Mike Burmester, Lightweight cryptographic mechanisms based on pseudorandom number generators 4. Philippe Cara, Loops, neardomains, nearfields and sets of permutations 5. Ilaria Cardinali, On the structure of Weyl modules for the symplectic group 6. Bill Cherowitzo, Parallelisms of quadrics 7. Jan De Beule, Large maximal partial ovoids of Q−(5, q) 8. Bart De Bruyn, On extensions of hyperplanes of dual polar spaces 1 9. Frank De Clerck, Intriguing sets of partial quadrangles 10. Alice Devillers, Symmetry properties of subdivision graphs 11. Dalibor Froncek, Decompositions of complete bipartite graphs into generalized prisms 12. Stelios Georgiou, Self-dual codes from circulant matrices 13. Robert Gilman, Cryptology of infinite groups 14. Otokar Groˇsek, The number of associative triples in a quasigroup 15. Christoph Hering, Latin squares, homologies and Euler’s conjecture 16. Leanne Holder, Bilinear star flocks of arbitrary cones 17. Robert Jajcay, On the geometry of cages 18.
  • Codes and Modules Associated with Designs and T-Uniform Hypergraphs Richard M

    Codes and Modules Associated with Designs and T-Uniform Hypergraphs Richard M

    Codes and modules associated with designs and t-uniform hypergraphs Richard M. Wilson California Institute of Technology (1) Smith and diagonal form (2) Solutions of linear equations in integers (3) Square incidence matrices (4) A chain of codes (5) Self-dual codes; Witt’s theorem (6) Symmetric and quasi-symmetric designs (7) The matrices of t-subsets versus k-subsets, or t-uniform hy- pergaphs (8) Null designs (trades) (9) A diagonal form for Nt (10) A zero-sum Ramsey-type problem (11) Diagonal forms for matrices arising from simple graphs 1. Smith and diagonal form Given an r by m integer matrix A, there exist unimodular matrices E and F ,ofordersr and m,sothatEAF = D where D is an r by m diagonal matrix. Here ‘diagonal’ means that the (i, j)-entry of D is 0 unless i = j; but D is not necessarily square. We call any matrix D that arises in this way a diagonal form for A. As a simple example, ⎛ ⎞ 0 −13 10 314⎜ ⎟ 100 ⎝1 −1 −1⎠ = . 21 4 −27 050 01−2 The matrix on the right is a diagonal form for the middle matrix on the left. Let the diagonal entries of D be d1,d2,d3,.... ⎛ ⎞ 20 0 ··· ⎜ ⎟ ⎜0240 ···⎟ ⎜ ⎟ ⎝0 0 120 ···⎠ . ... If all diagonal entries di are nonnegative and di divides di+1 for i =1, 2,...,thenD is called the integer Smith normal form of A, or simply the Smith form of A, and the integers di are called the invariant factors,ortheelementary divisors of A.TheSmith form is unique; the unimodular matrices E and F are not.
  • Modules Over Semisymmetric Quasigroups

    Modules Over Semisymmetric Quasigroups

    Modules over semisymmetric quasigroups Alex W. Nowak Abstract. The class of semisymmetric quasigroups is determined by the iden- tity (yx)y = x. We prove that the universal multiplication group of a semisym- metric quasigroup Q is free over its underlying set and then specify the point- stabilizers of an action of this free group on Q. A theorem of Smith indicates that Beck modules over semisymmetric quasigroups are equivalent to modules over a quotient of the integral group algebra of this stabilizer. Implementing our description of the quotient ring, we provide some examples of semisym- metric quasigroup extensions. Along the way, we provide an exposition of the quasigroup module theory in more general settings. 1. Introduction A quasigroup (Q, ·, /, \) is a set Q equipped with three binary operations: · denoting multiplication, / right division, and \ left division; these operations satisfy the identities (IL) y\(y · x)= x; (IR) x = (x · y)/y; (SL) y · (y\x)= x; (SR) x = (x/y) · y. Whenever possible, we convey multiplication of quasigroup elements by concatena- tion. As a class of algebras defined by identities, quasigroups constitute a variety (in the universal-algebraic sense), denoted by Q. Moreover, we have a category of quasigroups (also denoted Q), the morphisms of which are quasigroup homomor- phisms respecting the three operations. arXiv:1908.06364v1 [math.RA] 18 Aug 2019 If (G, ·) is a group, then defining x/y := x · y−1 and x\y := x−1 · y furnishes a quasigroup (G, ·, /, \). That is, all groups are quasigroups, and because of this fact, constructions in the representation theory of quasigroups often take their cue from familiar counterparts in group theory.
  • ON the POLYHEDRAL GEOMETRY of T–DESIGNS

    ON the POLYHEDRAL GEOMETRY of T–DESIGNS

    ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master of Arts In Mathematics by Steven Collazos San Francisco, California August 2013 Copyright by Steven Collazos 2013 CERTIFICATION OF APPROVAL I certify that I have read ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS by Steven Collazos and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University. Matthias Beck Associate Professor of Mathematics Felix Breuer Federico Ardila Associate Professor of Mathematics ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS Steven Collazos San Francisco State University 2013 Lisonek (2007) proved that the number of isomorphism types of t−(v; k; λ) designs, for fixed t, v, and k, is quasi{polynomial in λ. We attempt to describe a region in connection with this result. Specifically, we attempt to find a region F of Rd with d the following property: For every x 2 R , we have that jF \ Gxj = 1, where Gx denotes the G{orbit of x under the action of G. As an application, we argue that our construction could help lead to a new combinatorial reciprocity theorem for the quasi{polynomial counting isomorphism types of t − (v; k; λ) designs. I certify that the Abstract is a correct representation of the content of this thesis. Chair, Thesis Committee Date ACKNOWLEDGMENTS I thank my advisors, Dr. Matthias Beck and Dr.
  • Hadamard and Conference Matrices

    Hadamard and Conference Matrices

    Hadamard and conference matrices Peter J. Cameron December 2011 with input from Dennis Lin, Will Orrick and Gordon Royle Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI.
  • The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: Gasarch@Cs.Umd.Edu

    The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: [email protected]

    The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 email: [email protected] In the last issue of SIGACT NEWS (Volume 45, No. 3) there was a review of Martin Davis's book The Universal Computer. The road from Leibniz to Turing that was badly typeset. This was my fault. We reprint the review, with correct typesetting, in this column. In this column we review the following books. 1. The Universal Computer. The road from Leibniz to Turing by Martin Davis. Reviewed by Haim Kilov. This book has stories of the personal, social, and professional lives of Leibniz, Boole, Frege, Cantor, G¨odel,and Turing, and explains some of the essentials of their thought. The mathematics is accessible to a non-expert, although some mathematical maturity, as opposed to specific knowledge, certainly helps. 2. From Zero to Infinity by Constance Reid. Review by John Tucker Bane. This is a classic book on fairly elementary math that was reprinted in 2006. The author tells us interesting math and history about the numbers 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; e; @0. 3. The LLL Algorithm Edited by Phong Q. Nguyen and Brigitte Vall´ee. Review by Krishnan Narayanan. The LLL algorithm has had applications to both pure math and very applied math. This collection of articles on it highlights both. 4. Classic Papers in Combinatorics Edited by Ira Gessel and Gian-Carlo Rota. Re- view by Arya Mazumdar. This book collects together many of the classic papers in combinatorics, including those of Ramsey and Hales-Jewitt.
  • Combinatorial Designs

    Combinatorial Designs

    LECTURE NOTES COMBINATORIAL DESIGNS Mariusz Meszka AGH University of Science and Technology, Krak´ow,Poland [email protected] The roots of combinatorial design theory, date from the 18th and 19th centuries, may be found in statistical theory of experiments, geometry and recreational mathematics. De- sign theory rapidly developed in the second half of the twentieth century to an independent branch of combinatorics. It has deep interactions with graph theory, algebra, geometry and number theory, together with a wide range of applications in many other disciplines. Most of the problems are simple enough to explain even to non-mathematicians, yet the solutions usually involve innovative techniques as well as advanced tools and methods of other areas of mathematics. The most fundamental problems still remain unsolved. Balanced incomplete block designs A design (or combinatorial design, or block design) is a pair (V; B) such that V is a finite set and B is a collection of nonempty subsets of V . Elements in V are called points while subsets in B are called blocks. One of the most important classes of designs are balanced incomplete block designs. Definition 1. A balanced incomplete block design (BIBD) is a pair (V; B) where jV j = v and B is a collection of b blocks, each of cardinality k, such that each element of V is contained in exactly r blocks and any 2-element subset of V is contained in exactly λ blocks. The numbers v, b, r, k an λ are parameters of the BIBD. λ(v−1) λv(v−1) Since r = k−1 and b = k(k−1) must be integers, the following are obvious arithmetic necessary conditions for the existence of a BIBD(v; b; r; k; λ): (1) λ(v − 1) ≡ 0 (mod k − 1), (2) λv(v − 1) ≡ 0 (mod k(k − 1)).
  • One Method for Construction of Inverse Orthogonal Matrices

    One Method for Construction of Inverse Orthogonal Matrices

    ONE METHOD FOR CONSTRUCTION OF INVERSE ORTHOGONAL MATRICES, P. DIÞÃ National Institute of Physics and Nuclear Engineering, P.O. Box MG6, Bucharest, Romania E-mail: [email protected] Received September 16, 2008 An analytical procedure to obtain parametrisation of complex inverse orthogonal matrices starting from complex inverse orthogonal conference matrices is provided. These matrices, which depend on nonzero complex parameters, generalize the complex Hadamard matrices and have applications in spin models and digital signal processing. When the free complex parameters take values on the unit circle they transform into complex Hadamard matrices. 1. INTRODUCTION In a seminal paper [1] Sylvester defined a general class of orthogonal matrices named inverse orthogonal matrices and provided the first examples of what are nowadays called (complex) Hadamard matrices. A particular class of inverse orthogonal matrices Aa()ij are those matrices whose inverse is given 1 t by Aa(1ij ) (1 a ji ) , where t means transpose, and their entries aij satisfy the relation 1 AA nIn (1) where In is the n-dimensional unit matrix. When the entries aij take values on the unit circle A–1 coincides with the Hermitian conjugate A* of A, and in this case (1) is the definition of complex Hadamard matrices. Complex Hadamard matrices have applications in quantum information theory, several branches of combinatorics, digital signal processing, etc. Complex orthogonal matrices unexpectedly appeared in the description of topological invariants of knots and links, see e.g. Ref. [2]. They were called two- weight spin models being related to symmetric statistical Potts models. These matrices have been generalized to two-weight spin models, also called , Paper presented at the National Conference of Physics, September 10–13, 2008, Bucharest-Mãgurele, Romania.
  • Perfect Sequences Over the Quaternions and (4N, 2, 4N, 2N)-Relative Difference Sets in Cn × Q8

    Perfect Sequences Over the Quaternions and (4N, 2, 4N, 2N)-Relative Difference Sets in Cn × Q8

    Perfect Sequences over the Quaternions and (4n; 2; 4n; 2n)-Relative Difference Sets in Cn × Q8 Santiago Barrera Acevedo1? and Heiko Dietrich2? 1 Monash University, School of Mathematical Sciences, Clayton 3800 VIC, Australia [email protected] 2 Monash University, School of Mathematical Sciences, Clayton 3800 VIC, Australia [email protected] Abstract. Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±1; ±i; ±j; ±kg was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence be- tween perfect sequences of length n over Q8 [ qQ8 with q = (1 + i + j + k)=2, and (4n; 2; 4n; 2n)-relative difference sets in Cn × Q8 with forbidden subgroup a C2; here Cm is a cyclic group of order m. We show that if n = p + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n; 2; 4n; 2n)- relative different set in Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 [ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 [ qQ8). The final publication is available at Springer via http://link.springer.com/article/10.1007/s12095-017-0224-y 1 Introduction The periodic autocorrelation of a sequence is a measure for how much the sequence differs from its cyclic shifts. If the autocorrelation values for all nontrivial cyclic shifts are 0, then the sequence is perfect.
  • Hadamard Matrices Include

    Hadamard Matrices Include

    Hadamard and conference matrices Peter J. Cameron University of St Andrews & Queen Mary University of London Mathematics Study Group with input from Rosemary Bailey, Katarzyna Filipiak, Joachim Kunert, Dennis Lin, Augustyn Markiewicz, Will Orrick, Gordon Royle and many happy returns . Happy Birthday, MSG!! Happy Birthday, MSG!! and many happy returns . Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI.