Hadamard Matrices

Emeritus Professor Mieko Yamada of Kanazawa University graduated from Tokyo Woman's Christian University and received her PhD from Kyusyu University in 1987. She has taught at Tokyo Woman's Christian University, Konan University, Kyushu University, and Kanazawa University. Her areas of research are combinatorics, especially Hadamard matrices, difference sets and codes. Her research approach for combinatorics is based on number theory and algebra. She is a foundation fellow of Institute of Combinatorics and its Applications (ICA). She is an author of 51 papers in combinatorics and number theory. Emeritus Professor Jennifer Seberry graduated from University of New South Wales and received her PhD in Computation Mathematics from La Trobe University in 1971. She has held positions at the Australian National University, The University of Sydney, University College, The Australian Defence Force Academy (ADFA), The University of New South Wales, and University of Wollongong. She served as a head of Department of Computer Science of ADFA and a director of Centre for Computer Security Research of ADFA at University of Wollongong. She has published over 450 papers and eight books in Hadamard matrices, orthogonal designs, statistical designs, cryptology, and computer security.

• List of Tables xiiiList of Figures xvPreface xviiAcknowledgments xixAcronyms xxiIntroduction xxiii1 Basic Definitions 11.1 Notations 11.2 Finite Fields 11.3 Group Rings and Their Characters 81.4 Type 1 and Type 2 Matrices 91.5 Hadamard Matrices 141.6 Paley Core Matrices 201.7 Amicable Hadamard Matrices 221.8 The Additive Property and Four Plug-In Matrices 261.9 Difference Sets, Supplementary Difference Sets, and Partial Difference Sets 281.10 Sequences and Autocorrelation Function 331.11 Excess 371.12 Balanced Incomplete Block Designs 391.13 Hadamard Matrices and SBIBDs 411.14 Cyclotomic Numbers 411.15 Orthogonal Designs and Weighing Matrices 461.16 T-matrices, T-sequences, and Turyn Sequences 472 Gauss Sums, Jacobi Sums, and Relative Gauss Sums 492.1 Notations 492.2 Gauss Sums 492.3 Jacobi Sums 512.4 Cyclotomic Numbers and Jacobi Sums 602.5 Relative Gauss Sums 692.6 Prime Ideal Factorization of Gauss Sums 723 Plug-In Matrices 773.1 Notations 773.2 Williamson Type and Williamson Matrices 773.3 Plug-In Matrices 823.4 Eight Plug-In Matrices 843.5 More T-sequences and T-matrices 853.6 Construction of T-matrices of Order 6m + 1 873.7 Williamson Hadamard Matrices and Paley Type II Hadamard Matrices 903.8 Hadamard Matrices of Generalized Quaternion Type 973.9 Supplementary Difference Sets and Williamson Matrices 1003.10 Relative Difference Sets and Williamson-Type Matrices over Abelian Groups 1103.11 Computer Construction of Williamson Matrices 1124 Arrays: Matrices to Plug-Into 1154.1 Notations 1154.2 Orthogonal Designs 1154.3 Welch and Ono–Sawade–Yamamoto Arrays 1214.4 Regular Representation of a Group and BHW(G) 1225 Sequences 1255.1 Notations 1255.2 PAF and NPAF 1255.3 Suitable Single Sequences 1265.4 Suitable Pairs of NPAF Sequences: Golay Sequences 1315.5 Current Results for Golay Pairs 1315.6 Recent Results for Periodic Golay Pairs 1335.7 More on Four Complementary Sequences 1335.8 6-Turyn-Type Sequences 1365.9 Base Sequences 1375.10 Yang-Sequences 1376 M-structures 1456.1 Notations 1456.2 The Strong Kronecker Product 1456.3 Reducing the Powers of 2 1476.4 Multiplication Theorems Using M-structures 1496.5 Miyamoto's Theorem and Corollaries via M-structures 1517 Menon Hadamard Difference Sets and Regular Hadamard Matrices 1597.1 Notations 1597.2 Menon Hadamard Difference Sets and Exponent Bound 1597.3 Menon Hadamard Difference Sets and Regular Hadamard Matrices 1607.4 The Constructions from Cyclotomy 1617.5 The Constructions Using Projective Sets 165x Contents7.6 The Construction Based on Galois Rings 1708 Paley Hadamard Difference Sets and Paley Type Partial Difference Sets 1758.1 Notations 1758.2 Paley Core Matrices and Gauss Sums 1758.3 Paley Hadamard Difference Sets 1788.4 Paley Type Partial Difference Set 1828.5 The Construction of Paley Type PDS from a Covering Extended Building Set 1838.6 Constructing Paley Hadamard Difference Sets 1919 Skew Hadamard, Amicable, and Symmetric Matrices 1939.1 Notations 1939.2 Introduction 1939.3 Skew Hadamard Matrices 1939.4 Constructions for Skew Hadamard Matrices 1959.5 Szekeres Difference Sets 2009.6 Amicable Hadamard Matrices 2049.7 Amicable Cores 2079.8 Construction for Amicable Hadamard Matrices of Order 2t 2089.9 Construction of Amicable Hadamard Matrices Using Cores 2099.10 Symmetric Hadamard Matrices 21110 Skew Hadamard Difference Sets 21510.1 Notations 21510.2 Skew Hadamard Difference Sets 21510.3 The Construction by Planar Functions Over a Finite Field 21510.4 The Construction by Using Index 2 Gauss Sums 21810.5 The Construction by Using Normalized Relative Gauss Sums 22611 Asymptotic Existence of Hadamard Matrices 23311.1 Notations 23311.2 Introduction 23311.3 Seberry's Theorem 23311.4 Craigen's Theorem 23411.5 More Asymptotic Theorems 24311.6 Skew Hadamard and Regular Hadamard 24312 More on Maximal Determinant Matrices 24512.1 Notations 24512.2 E-Equivalence: The Smith Normal Form 24512.3 E-Equivalence: The Number of Small Invariants 24712.4 E-Equivalence: Skew Hadamard and Symmetric Conference Matrices 25012.5 Smith Normal Form for Powers of 2 25212.6 Matrices with Elements (1, −1) and Maximal Determinant 25312.7 D-Optimal Matrices Embedded in Hadamard Matrices 25412.8 Embedding of Hadamard Matrices within Hadamard Matrices 25712.9 Embedding Properties Via Minors 25712.10 Embeddability of Hadamard Matrices 25912.11 Embeddability of Hadamard Matrices of Order n − 8 26012.12 Embeddability of Hadamard Matrices of Order n − k 26112.13 Growth Problem for Hadamard Matrices 265A Hadamard Matrices 271B List of sds from Cyclotomy 295C Further Research Questions 301References 303Index 313