Circulant Weighing Matrices
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Second Edition Volume I
Second Edition Thomas Beth Universitat¨ Karlsruhe Dieter Jungnickel Universitat¨ Augsburg Hanfried Lenz Freie Universitat¨ Berlin Volume I PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain First edition c Bibliographisches Institut, Zurich, 1985 c Cambridge University Press, 1993 Second edition c Cambridge University Press, 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in LATEX2ε[TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949– Design theory / Thomas Beth, Dieter Jungnickel, Hanfried Lenz. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 44432 2 (hardbound) 1. Combinatorial designs and configurations. I. Jungnickel, D. (Dieter), 1952– . II. Lenz, Hanfried. III. Title. QA166.25.B47 1999 5110.6 – dc21 98-29508 CIP ISBN 0 521 44432 2 hardback Contents I. Examples and basic definitions .................... 1 §1. Incidence structures and incidence matrices ............ 1 §2. Block designs and examples from affine and projective geometry ........................6 §3. t-designs, Steiner systems and configurations ......... -
Matrix Algebra and Control
Appendix A Matrix Algebra and Control Boldface lower case letters, e.g., a or b, denote vectors, boldface capital letters, e.g., A, M, denote matrices. A vector is a column matrix. Containing m elements (entries) it is referred to as an m-vector. The number of rows and columns of a matrix A is nand m, respectively. Then, A is an (n, m)-matrix or n x m-matrix (dimension n x m). The matrix A is called positive or non-negative if A>, 0 or A :2:, 0 , respectively, i.e., if the elements are real, positive and non-negative, respectively. A.1 Matrix Multiplication Two matrices A and B may only be multiplied, C = AB , if they are conformable. A has size n x m, B m x r, C n x r. Two matrices are conformable for multiplication if the number m of columns of the first matrix A equals the number m of rows of the second matrix B. Kronecker matrix products do not require conformable multiplicands. The elements or entries of the matrices are related as follows Cij = 2::;;'=1 AivBvj 'Vi = 1 ... n, j = 1 ... r . The jth column vector C,j of the matrix C as denoted in Eq.(A.I3) can be calculated from the columns Av and the entries BVj by the following relation; the jth row C j ' from rows Bv. and Ajv: column C j = L A,vBvj , row Cj ' = (CT),j = LAjvBv, (A. 1) /1=1 11=1 A matrix product, e.g., AB = (c: c:) (~b ~b) = 0 , may be zero although neither multipli cand A nor multiplicator B is zero. -
LCD Codes from Weighing Matrices
LCD codes from weighing matrices Dean Crnkovi´c, Ronan Egan, B. G. Rodrigues and Andrea Svobˇ Abstract Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order q using weighing matrices and their orbit matrices. The LCD codes constructed can be of any length dimension according to the choice of matrices used in their construction. As a special case, LCD codes of length 2n and dimension n are constructed which also have the property of being formally self-dual. Alternatively, under a condition depending on q that the codes are not LCD, this method constructs self-dual codes. To illustrate the method we construct LCD codes from weighing matrices, including the Paley conference matrices and Hadamard matrices. We also extend the construction to Hermitian LCD codes over the finite field of order 4. In addition, we propose a decoding algorithm that can be feasible for the LCD codes obtained from some of the given methods. 2010 Mathematics Subject Classification: 05B20, 05B30, 94B05, 12E20. Keywords: Weighing matrix, orbit matrix, LCD code. 1 Introduction In this paper we introduce a method for constructing linear codes with complementary duals, or LCD codes. LCD codes are linear codes whose intersection with their duals are trivial; they were introduced by Massey in [23] and have been widely applied in information protection, arXiv:1812.00368v2 [math.CO] 28 Nov 2019 electronics and cryptography. They provide an optimum linear coding solution for the two user binary adder channel. -
Balanced Generalized Weighing Matrices and Their Applications
LE MATEMATICHE Vol. LIX (2004) – Fasc. I–II, pp. 225–261 BALANCED GENERALIZED WEIGHING MATRICES AND THEIR APPLICATIONS DIETER JUNGNICKEL - H. KHARAGHANI Balanced generalized weighing matrices include well-known classical combinatorial objects such as Hadamard matrices and conference matrices; moreover, particular classes of BGW-matrices are equivalent to certain rel- ative difference sets. BGW-matrices admit an interesting geometrical inter- pretation, and in this context they generalize notions like projective planes admitting a full elation or homology group. After surveying these basic con- nections, we will focus attention on proper BGW-matrices; thus we will not give any systematic treatment of generalized Hadamard matrices, which are the subject of a large area of research in their own right. In particular, we will discuss what might be called the classical param- eter series. Here the nicest examples are closely related to perfect codes and to some classical relative difference sets associated with affine geometries; moreover, the matrices in question can be characterized as the unique (up to equivalence) BGW-matrices for the given parameters with minimum q-rank. One can also obtain a wealth of monomially inequivalent examples and deter- mine the q-ranks of all these matrices by exploiting a connection with linear shift register sequences. The final section of this survey will consider applications to construc- tions of designs and graphs. We will divide this section into six parts. The first of these will deal with the work of Ionin. The second part will be devoted to Bush–type Hadamard matrices and twin designs. In the third part we will Keywords: Balanced generalized weighing matrix, Relative difference set, Simplex code, Symmetric design, Regular Hadamard matrix, Productive regular Hadamard matrix, Bush–type Hadamard matrix 226 DIETER JUNGNICKEL - H. -
Orr Hunter K 202104 Msc.Pdf (5.035Mb)
Modelling Human Behaviour Based on Similarity Measurements Between Event Sequences by Hunter Orr A thesis submitted to the School of Computing in conformity with the requirements for the degree of Master of Science Queen's University Kingston, Ontario, Canada May 2021 Copyright c Hunter Orr, 2021 Abstract From a set of sequences, individual's behavioural patterns can be identified. Using these sequences of events, the metadata available can be processed into a weighted format to improve the meaningfulness of the sequence comparisons. The usefulness of this process, identifying users' behavioural patterns, is important in a number of areas such as cybersecurity. This work examines the properties a cybersecurity dataset might contain and demonstrates its effectiveness on a dataset with those properties. Building on the existing sequence comparison method, Damerau-levenshtein dis- tance, this work develops a pipeline of steps that can be used to transform the meta- data and integrate this weighted format into the sequence comparison calculation. In this pipeline, one of the most significant transformations that is applied to the meta- data is based on previous work by Brand. This transformation reduces the impact of high popularity pairwise relationships. This pipeline is shown to incorporate the metadata information into the resulting distance values. Thus, producing meaningful changes which demonstrate the benefit of these extra steps. i Acknowledgments I would like to thank Dr. David Skillicorn for his guidance and thoughtful discussions throughout this entire work. The discussions we had were invaluable and continue to encourage me to improve my thinking process. I appreciate the enthusiasm and encouragement you have provided throughout this research. -
Trace, Symmetry and Orthogonality
Canad. Math. Bull. Vol. 37 (4), 1994 pp. 461-467 TRACE, SYMMETRY AND ORTHOGONALITY R. CRAIGEN ABSTRACT. Does there exist a circulant conference matrix of order > 2? When is there a symmetric Hadamard matrix with constant diagonal? How many pairwise disjoint, amicable weighing matrices of order n can there be? These are questions concerning which the trace function gives a great deal of insight. We offer easy proofs of the known solutions to the first two, the first being new, and develop new results regarding the latter question. It is shown that there are 2* disjoint amicable weighing matrices of order 2*p, where p is odd, and that this is an upper bound for t < 1. An even stronger bound is obtained for certain cases. 1. Introduction and preliminaries. Weighing matrices A, B are said to be amicable if ABl = BAl (and antiamicable if ABf + BAl = 0). Clearly A and B are amicable (antiamicable) if and only if At and Bl are as well. We denote by A C\B the Hadamard, or entry-wise product of matrices A, B having the same size. These are disjoint if A HB = 0. One of the most powerful means by which Hadamard matrices are constructed is with orthogonal designs (see [7]). An orthogonal design OD{n\s\,... ,ty) corresponds to k pairwise disjoint, antiamicable weighing matrices of order n, with weights si,... ,^. Similar constructions use disjoint, amicable weighing matrices (see [3, Theorem 4.2], [5]). Even without the disjointness property, amicable weighing matrices are of consid erable interest (see pp. 217-219 of [7]). -
Discrete Mathematics Generalized Weighing Matrices and Self
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 309 (2009) 4697–4699 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Note Generalized weighing matrices and self-orthogonal codes Vladimir D. Tonchev Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA article info a b s t r a c t Article history: It is shown that the row spaces of certain generalized weighing matrices, Bhaskar Rao Received 11 October 2006 designs, and generalized Hadamard matrices over a finite field of order q are hermitian Accepted 28 May 2008 self-orthogonal codes. Certain matrices of minimum rank yield optimal codes. In the special Available online 1 July 2008 case when q D 4, the codes are linked to quantum error-correcting codes, including some codes with optimal parameters. Keywords: ' 2008 Elsevier B.V. All rights reserved. Generalized weighing matrix Generalized Hadamard matrix Bhaskar Rao design Self-orthogonal code Quantum code 1. Introduction A generalized weighing matrix over a multiplicative group G of order g is a v × b matrix M D .mij/ with entries from G [ f0g such that for every two rows .mi1;:::; mib/; .mj1;:::; mjb/, i 6D j, the multi-set f −1 j ≤ ≤ 6D g mismjs 1 s b; mjs 0 (1) contains every element of G the same number of times. A generalized weighing matrix with the additional properties that every row contains precisely r nonzero entries, each column contains exactly k nonzero entries, and for every two distinct rows the multi-set (1) contains every group element exactly λ/g times is known as a generalized Bhaskar Rao design GBRD.v; b; r; k; λI G/ [10]. -
Applications of Finite Geometries to Designs and Codes
Michigan Technological University Digital Commons @ Michigan Tech Dissertations, Master's Theses and Master's Dissertations, Master's Theses and Master's Reports - Open Reports 2011 Applications of finite geometries ot designs and codes David C. Clark Michigan Technological University Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons Copyright 2011 David C. Clark Recommended Citation Clark, David C., "Applications of finite geometries ot designs and codes", Dissertation, Michigan Technological University, 2011. https://doi.org/10.37099/mtu.dc.etds/199 Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons APPLICATIONS OF FINITE GEOMETRIES TO DESIGNS AND CODES By David C. Clark A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Mathematical Sciences) MICHIGAN TECHNOLOGICAL UNIVERSITY 2011 c 2011 David C. Clark This dissertation, "Applications of Finite Geometries to Designs and Codes," is hereby approved in partial fulfillment of the requirements for the Degree of DOC- TOR OF PHILOSOPHY IN MATHEMATICAL SCIENCES. Department of Mathematical Sciences Signatures: Dissertation Advisor Dr. Vladimir Tonchev Committee Member Dr. Donald Kreher Committee Member Dr. Stefaan De Winter Committee Member Dr. Steven Seidel Department Chair Dr. Mark Gockenbach Date Contents List of Figures ..................................... ix List of Tables ...................................... xi Preface .........................................xiii Acknowledgments ................................... xv Abstract .........................................xvii 1 Designs, codes, and finite geometries ....................... 1 1.1 Introduction .................................. 1 1.1.1 Designs ................................ 2 1.1.2 Error-correcting codes ........................ 4 1.1.3 The codes of a design and p-ranks .................. 5 1.2 Finite geometry designs and codes ..................... -
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Applying balanced generalized weighing matrices to construct block designs Yury J. Ionin Department of Mathematics Central Michigan University Mt. Pleasant, MI 48858, USA [email protected] Submitted: October 10, 2000; Accepted: January 27, 2001. MR Subject Classifications: 05B05 Abstract Balanced generalized weighing matrices are applied for constructing a family of symmetric designs with parameters (1 + qr(rm+1 − 1)/(r − 1),rm,rm−1(r − 1)/q), where m is any positive integer and q and r =(qd − 1)/(q − 1) are prime powers, and a family of non-embeddable quasi-residual 2−((r +1)(rm+1 −1)/(r −1),rm(r + 1)/2,rm(r − 1)/2) designs, where m is any positive integer and r =2d − 1, 3 · 2d − 1 or 5 · 2d − 1 is a prime power, r ≥ 11. 1 Introduction A balanced incomplete block design (BIBD) with parameters (v, b, r,k, λ)ora2-(v, k, λ) design is a pair D =(V,B), where V is a set (of points) of cardinality v and B is a collection of bk-subsets of V (blocks) such that each point is contained in exactly r blocks and each 2-subset of V is contained in exactly λ blocks. If V = {x1,x2,... ,xv} and B = {B1,B2,... ,Bb}, then the v ×b matrix, whose (i, j)-entry is equal to 1 if xi ∈ Bj and is equal to 0 otherwise, is the incidence matrix of the design. A (0, 1) matrix X of size t v × b is the incidence matrix of a (v, b, r,k, λ) BIBD if and only if XX =(r − λ)Iv + λJv and JvX = kJv×b,whereIv, Jv,andJv×b are the identity matrix of order v,thev × v all-one matrix, and the v × b all-one matrix, respectively. -
On Generalized Hadamard Matrices and Difference Matrices: $ Z 6$
On Generalized Hadamard Matrices and Difference Matrices: Z6 Bradley W. Brock∗, Robert Compton†, Warwick de Launey‡ and Jennifer Seberry§ October 26, 2015 We give some very interesting matrices which are orthogonal over groups and, as far as we know, referenced, but in fact undocumented. This note is not intended to be published but available for archival reasons. Keywords: Difference Matrices, Generalised/Generalized Hadamard Matrices; Bhaskar Rao Designs, Orthogonal Matrices; Cretan Matrices; Butson-Hadamard Matrices; 05B20. 1 Introduction (Caution: please note Jennifer Seberry has tried but been unable to contact the persons named as co-authors. Apolgogies for half finished results.) We note that the literature on this subject is very disorganized. Authors have not read the literature on their own key-words and papers thus ignored, on the other hand papers have been claimed to be published which have not. We have put together this note to make them accessible to all. We do not give examples of all the literature but point to references which may help. This work is compiled from journal work of the authors from old pieces of paper. We first note that the matrices we study here have elements from groups, abelian and non-abelian, and may be written in additive or multiplicative notation. The matrices may have real elements, elements ∈ {1, −1}, elements SnS≤ 1, elements ∈ {1, i, i2 = −1}, elements ∈ {1, i, −1, −i, i2 =−1}, integer elements ∈ {a + ib, i2 =−1}, nth roots of unity, the quaternions {1 and i, j, k, i2 = j2 = k2 =−1, ijk =−1}, (a + ib)+ j(c + id), a, b, c, d, arXiv:1510.06816v1 [math.CO] 23 Oct 2015 integer and i, j, k quaternions or otherwise as specified. -
Some Non-Existence and Asymptotic Existence Results for Weighing
Some non-existence and asymptotic existence results for weighing matrices Ebrahim Ghaderpour 1 Lassonde School of Engineering York University, Canada Abstract Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking and communication. In this paper, we first show that if positive integer k cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order 4n and weight k, where n is an odd positive integer. Then we show that for any square k, there is an integer N(k) such that for each n N(k), there is a symmetric weighing matrix of order n and weight k.≥ Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita and Seberry. Keywords: Asymptotic existence; Orthogonal design; Skew-symmetric weighing matrix; Symmetric weighing matrix; Weighing matrix. 1 Introduction An orthogonal design (OD) [3, Chapter 1] of order n and type (s1,...,sℓ), arXiv:1602.07794v1 [math.CO] 23 Feb 2016 denoted OD(n; s1,...,sℓ), is a square matrix X of order n with entries from 0, x ,..., xℓ , where the xj’s are commuting variables, that satisfies { ± 1 ± } ℓ T 2 XX = sjxj In, j ! X=1 T where X is the transpose of X, and In is the identity matrix of order n. An OD with no zero entry is called a full OD. Equating all variables to 1 1E-mail: [email protected]. 1 in any full OD results in a Hadamard matrix. Equating all variables to 1 in any OD of order n results a weighing matrix, denoted W (n, k), where k is the weight that is the number of nonzero entries in each row (column) of the weighing matrix. -
On Cocyclic Weighing Matrices and the Regular Group Actions of Certain
Discrete Applied Mathematics 102 (2000) 63{101 View metadata, citation and similar papers at core.ac.uk brought to you by CORE On cocyclic weighing matrices and the regular group provided by Elsevier - Publisher Connector actions of certain paley matrices Warwick de Launey∗, Richard M. Sta ord Center for Communications Research, 4320 Westerra Court, San Diego, California, CA92121, USA Received 1 September 1998; revised 7 June 1999; accepted 2 August 1999 Abstract In this paper we consider cocyclic weighing matrices. Cocyclic development of a weighing matrix is shown to be related to regular group actions on the points of the associated group divisible design. We show that a cocyclic weighing matrix is equivalent to a relative di erence set with central forbidden subgroup of order two. We then set out an agenda for studying a known cocyclic weighing matrix and carry it out for the Paley conference matrix and for the type I Paley Hadamard matrix. Using a connection with certain near ÿelds, we determine all the regular group actions on the group divisible design associated to such a Paley matrix. It happens that all the regular actions of the Paley type I Hadamard matrix have already been described in the literature, however, new regular actions are identiÿed for the Paley conference matrix. This allows us to determine all the extension groups and indexing groups for the cocycles of the aforementioned Paley matrices, and gives new families of normal and non-normal relative di erence sets with forbidden subgroup of size two. ? 2000 Elsevier Science B.V. All rights reserved.