Up-to-date resource on Hadamard matrices

Hadamard Matrices: Constructions using Number Theory and Algebraprovides students with a discussion of the basic definitions used for Hadamard Matrices as well as more advanced topics in the subject, including:

The book can be used as a textbook for graduate courses in combinatorics, or as a reference for researchers studying Hadamard matrices.

Utilized in the fields of signal processing and design experiments, Hadamard matrices have been used for 150 years, and remain practical today.Hadamard Matricescombines a thorough discussion of the basic concepts underlying the subject matter with more advanced applications that will be of interest to experts in the area.

Emeritus ProfessorMieko 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 ProfessorJennifer 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 xiii

List of Figures xv

Preface xvii

Acknowledgments xix

Acronyms xxi

Introduction xxiii

1 Basic Definitions1

1.1 Notations 1

1.2 Finite Fields 1

1.2.1 A Residue Class Ring 1

1.2.2 Properties of Finite Fields 4

1.2.3 Traces and Norms 4

1.2.4 Characters of Finite Fields 6

1.3 Group Rings and Their Characters 8

1.4 Type 1 and Type 2 Matrices 9

1.5 Hadamard Matrices 14

1.5.1 Definition and Properties of an Hadamard Matrix 14

1.5.2 Kronecker Product and the Sylvester Hadamard Matrices 17

1.5.2.1 Remarks on Sylvester Hadamard Matrices 18

1.5.3 Inequivalence Classes 19

1.6 Paley Core Matrices 20

1.7 Amicable Hadamard Matrices 22

1.8 The Additive Property and Four Plug-In Matrices 26

1.8.1 Computer Construction 26

1.8.2 Skew Hadamard Matrices 27

1.8.3 Symmetric Hadamard Matrices 27

1.9 Difference Sets, Supplementary Difference Sets, and Partial Difference Sets 28

1.9.1 Difference Sets 28

1.9.2 Supplementary Difference Sets 30

1.9.3 Partial Difference Sets 31

1.10 Sequences and Autocorrelation Function 33

1.10.1 Multiplication of NPAF Sequences 35

1.10.2 Golay Sequences 36

1.11 Excess 37

1.12 Balanced Incomplete Block Designs 39

1.13 Hadamard Matrices and SBIBDs 41

1.14 Cyclotomic Numbers 41

1.15 Orthogonal Designs and Weighing Matrices 46

1.16T-matrices,T-sequences, and Turyn Sequences 47

1.16.1 Turyn Sequences 48

2 Gauss Sums, Jacobi Sums, and Relative Gauss Sums49

2.1 Notations 49

2.2 Gauss Sums 49

2.3 Jacobi Sums 51

2.3.1 Congruence Relations 52

2.3.2 Jacobi Sums of Order 4 52

2.3.3 Jacobi Sums of Order 8 57

2.4 Cyclotomic Numbers and Jacobi Sums 60

2.4.1 Cyclotomic Numbers fore= 2 62

2.4.2 Cyclotomic Numbers fore= 4 63

2.4.3 Cyclotomic Numbers fore= 8 64

2.5 Relative Gauss Sums 69

2.6 Prime Ideal Factorization of Gauss Sums 72

2.6.1 Prime Ideal Factorization of a Primep72

2.6.2 Stickelbergers Theorem 72

2.6.3 Prime Ideal Factorization of the Gauss Sum inQ(𝜁q1) 73

2.6.4 Prime Ideal Factorization of the Gauss Sums inQ(𝜁m) 74

3 Plug-In Matrices77

3.1 Notations 77

3.2 Williamson Type and Williamson Matrices 77

3.3 Plug-In Matrices 82

3.3.1 The Ito Array 82

3.3.2 Good Matrices : A Variation of Williamson Matrices 82

3.3.3 The GoethalsSeidel Array 83

3.3.4 Symmetric Hadamard Variation 84

3.4 Eight Plug-In Matrices 84

3.4.1 The Kharaghani Array 84

3.5 MoreT-sequences andT-matrices 85

3.6 Construction ofT-matrices of Order 6m+ 1 87

3.7 Williamson Hadamard Matrices and Paley Type II Hadamard Matrices 90

3.7.1 Whitemans Construction 90

3.7.2 Williamson Equation from Relative Gauss Sums 94

3.8 Hadamard Matrices of Generalized Quaternion Type 97

3.8.1 Definitions 97

3.8.2 Paley Core Type I Matrices 99

3.8.3 Infinite Families of Hadamard Matrices of GQ Type and Relative Gauss Sums 99

3.9 Supplementary Difference Sets and Williamson Matrices 100

3.9.1 Supplementary Difference Sets from Cyclotomic Classes 100

3.9.2 Constructions of an Hadamard 4-sds 102

3.9.3 Construction from (q;x, y)-Partitions 105

3.10 Relative Difference Sets and Williamson-Type Matrices over Abelian Groups 110

3.11 Computer Construction of Williamson Matrices 112

4 Arrays: Matrices to Plug-Into115

4.1 Notations 115

4.2 Orthogonal Designs 115

4.2.1 BaumertHall Arrays and Welch Arrays 116

4.3 Welch and OnoSawadeYamamoto Arrays 121

4.4 Regular Representation of a Group andBHW(G) 122

5 Sequences125

5.1 Notations 125

5.2 PAF and NPAF 125

5.3 Suitable Single Sequences 126

5.3.1 Thoughts on the Nonexistence of Circulant Hadamard Matrices for Orders>4 126

5.3.2 SBIBD Implications 127

5.3.3 From ±1 Matrices to ±A,±BMatrices 127

5.3.4 Matrix Specifics 129

5.3.5 Counting Two Ways 129

5.3.6 FormOdd: Orthogonal Design Implications 130

5.3.7 The Case for Order 16 130

5.4 Suitable Pairs of NPAF Sequences: Golay Sequences 131

5.5 Current Results for Golay Pairs 131

5.6 Recent Results for Periodic Golay Pairs 133

5.7 More on Four Complementary Sequences 133

5.8 6-Turyn-Type Sequences 136

5.9 Base Sequences 137

5.10 Yang-Sequences 137

5.10.1 On Yangs Theorems onT-sequences 140

5.10.2 Multiplying by 2g+ 1,gthe Length of a Golay Sequence 142

5.10.3 Multiplying by 7 and 13 143

5.10.4 Koukouvinos and Kounias Number 144

6M-structures145

6.1 Notations 145

6.2 The Strong Kronecker Product 145

6.3 Reducing the Powers of 2 147

6.4 Multiplication Theorems UsingM-structures 149

6.5 Miyamotos Theorem and Corollaries viaM-structures 151

7 Menon Hadamard Difference Sets and Regular Hadamard Matrices159

7.1 Notations 159

7.2 Menon Hadamard Difference Sets and Exponent Bound 159

7.3 Menon Hadamard Difference Sets and Regular Hadamard Matrices 160

7.4 The Constructions from Cyclotomy 161

7.5 The Constructions Using Projective Sets 165

7.5.1 Graphical Hadamard Matrices 169

7.6 The Construction Based on Galois Rings 170

7.6.1 Galois Rings 170

7.6.2 Additive Characters of Galois Rings 170

7.6.3 A New Operation 171

7.6.4 Gauss Sums OverGR(2n+1, s) 171

7.6.5 Menon Hadamard Difference Sets OverGR(2n+1, s) 172

7.6.6 Menon Hadamard Difference Sets OverGR(22, s) 173

8 Paley Hadamard Difference Sets and Paley Type Partial Difference Sets175

8.1 Notations 175

8.2 Paley Core Matrices and Gauss Sums 175

8.3 Paley Hadamard Difference Sets 178

8.3.1 StantonSprott Difference Sets 179

8.3.2 Paley Hadamard Difference Sets Obtained from Relative Gauss Sums 180

8.3.3 GordonMillsWelch Extension 181

8.4 Paley Type Partial Difference Set 182

8.5 The Construction of Paley Type PDS from a Covering Extended Building Set 183

8.6 Constructing Paley Hadamard Difference Sets 191

9 Skew Hadamard, Amicable, and Symmetric Matrices193

9.1 Notations 193

9.2 Introduction 193

9.3 Skew Hadamard Matrices 193

9.3.1 Summary of Skew Hadamard Orders 194

9.4 Constructions for Skew Hadamard Matrices 195

9.4.1 The GoethalsSeidel Type 196

9.4.2 An Adaption of WallisWhiteman Array 197

9.5 Szekeres Difference Sets 200

9.5.1 The Construction by Cyclotomic Numbers 202

9.6 Amicable Hadamard Matrices 204

9.7 Amicable Cores 207

9.8 Construction for Amicable Hadamard Matrices of Order 2t208

9.9 Construction of Amicable Hadamard Matrices Using Cores 209

9.10 Symmetric Hadamard Matrices 211

9.10.1 Symmetric Hadamard Matrices Via Computer Construction 212

9.10.2 Luchshie Matrices Known Results 212

10 Skew Hadamard Difference Sets215

10.1 Notations 215

10.2 Skew Hadamard Difference Sets 215

10.3 The Construction by Planar Functions Over a Finite Field 215

10.3.1 Planar Functions and Dickson Polynomials 215

10.4 The Construction by Using Index 2 Gauss Sums 218

10.4.1 Index 2 Gauss Sums 218

10.4.2 The Case thatp1 7 (mod 8) 219

10.4.3 The Case thatp1 3 (mod 8) 221

10.5 The Construction by Using Normalized Relative Gauss Sums 226

10.5.1 More on Ideal Factorization of the Gauss Sum 226

10.5.2 Determination of Normalized Relative Gauss Sums 226

10.5.3 A Family of Skew Hadamard Difference Sets 228

11 Asymptotic Existence of Hadamard Matrices233

11.1 Notations 233

11.2 Introduction 233

11.2.1 de Launeys Theorem 233

11.3 Seberrys Theorem 233

11.4 Craigens Theorem 234

11.4.1 Signed Groups and Their Representations 234

11.4.2 A Construction for Signed Group Hadamard Matrices 236

11.4.3 A Construction for Hadamard Matrices 238

11.4.4 Comments on Orthogonal Matrices Over Signed Groups 240

11.4.5 Some Calculations 241

11.5 More Asymptotic Theorems 243

11.6 Skew Hadamard and Regular Hadamard 243

12 More on Maximal Determinant Matrices245

12.1 Notations 245

12.2E-Equivalence: The Smith Normal Form 245

12.3E-Equivalence: The Number of Small Invariants 247

12.4E-Equivalence: Skew Hadamard and Symmetric Conference Matrices 250

12.5 Smith Normal Form for Powers of 2 252

12.6 Matrices with Elements (1,1) and Maximal Determinant 253

12.7D-Optimal Matrices Embedded in Hadamard Matrices 254

12.7.1 Embedding ofD5 inH8 254

12.7.2 Embedding ofD6 inH8 255

12.7.3 Embedding ofD7 inH8 255

12.7.4 Other Embeddings 255

12.8 Embedding of Hadamard Matrices within Hadamard Matrices 257

12.9 Embedding Properties Via Minors 257

12.10 Embeddability of Hadamard Matrices 259

12.11 Embeddability of Hadamard Matrices of Ordern 8 260

12.12 Embeddability of Hadamard Matrices of Ordernk261

12.12.1 EmbeddabilityExtendability of Hadamard Matrices 262

12.12.2 Available Determinant Spectrum and Verification 263

12.13 Growth Problem for Hadamard Matrices 265

A Hadamard Matrices271

A.1 Hadamard Matrices 271

A.1.1 Amicable Hadamard Matrices 271

A.1.2 Skew Hadamard Matrices 271

A.1.3 Spence Hadamard Matrices 272

A.1.4 Conference Matrices Give Symmetric Hadamard Matrices 272

A.1.5 Hadamard Matrices from Williamson Matrices 273

A.1.6 OD Hadamard Matrices 273

A.1.7 Yamada Hadamard Matrices 273

A.1.8 Miyamoto Hadamard Matrices 273

A.1.9 Koukouvinos and Kounias 273

A.1.10 Yang Numbers 274

A.1.11 Agaian Multiplication 274

A.1.12 CraigenSeberryZhang 274

A.1.13 de Launey 274

A.1.14 Seberry/Craigen Asymptotic Theorems 275

A.1.15 Yangs Theorems and Dokovic Updates 275

A.1.16 Computation by Dokovic 275

A.2 Index of Williamson Matrices 275

A.3 Tables of Hadamard Matrices 276

B List of sds from Cyclotomy295

B.1 Introduction 295

B.2 List ofn {q;k1,, kn𝜆}sds295

C Further Research Questions301

C.1 Research Questions for Future Investigation 301

C.1.1 Matrices 301

C.1.2 Base Sequences 301

C.1.3 Partial Difference Sets 301

C.1.4 de Launeys Four Questions 301

C.1.5 Embedding Sub-matrices 302

C.1.6 Pivot Structures 302

C.1.7 Trimming and Bordering 302

C.1.8 Arrays 302

References 303

Index 313