1 Basic Definitions 1
1.1 Notations 1
1.2 Finite Fields 1
1.3 Group Rings and Their Characters 8
1.4 Type 1 and Type 2 Matrices 9
1.5 Hadamard Matrices 14
1.6 Paley Core Matrices 20
1.7 Amicable Hadamard Matrices 22
1.8 The Additive Property and Four Plug-In Matrices 26
1.9 Difference Sets, Supplementary Difference Sets, and Partial Difference Sets 28
1.10 Sequences and Autocorrelation Function 33
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
1.16 T-matrices, T-sequences, and Turyn Sequences 47

2 Gauss Sums, Jacobi Sums, and Relative Gauss Sums 49
2.1 Notations 49
2.2 Gauss Sums 49
2.3 Jacobi Sums 51
2.4 Cyclotomic Numbers and Jacobi Sums 60
2.5 Relative Gauss Sums 69
2.6 Prime Ideal Factorization of Gauss Sums 72

3 Plug-In Matrices 77
3.1 Notations 77
3.2 Williamson Type and Williamson Matrices 77
3.3 Plug-In Matrices 82
3.4 Eight Plug-In Matrices 84
3.5 More T-sequences and T-matrices 85
3.6 Construction of T-matrices of Order 6m + 1 87
3.7 Williamson Hadamard Matrices and Paley Type II Hadamard Matrices 90
3.8 Hadamard Matrices of Generalized Quaternion Type 97
3.9 Supplementary Difference Sets and Williamson Matrices 100
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-Into 115
4.1 Notations 115
4.2 Orthogonal Designs 115
4.3 Welch and Ono–Sawade–Yamamoto Arrays 121
4.4 Regular Representation of a Group and BHW(G) 122

5 Sequences 125
5.1 Notations 125
5.2 PAF and NPAF 125
5.3 Suitable Single Sequences 126
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

6 M-structures 145
6.1 Notations 145
6.2 The Strong Kronecker Product 145
6.3 Reducing the Powers of 2 147
6.4 Multiplication Theorems Using M-structures 149
6.5 Miyamoto's Theorem and Corollaries via M-structures 151

7 Menon Hadamard Difference Sets and Regular Hadamard Matrices 159
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 165x Contents
7.6 The Construction Based on Galois Rings 170

8 Paley Hadamard Difference Sets and Paley Type Partial Difference Sets 175
8.1 Notations 175
8.2 Paley Core Matrices and Gauss Sums 175
8.3 Paley Hadamard Difference Sets 178
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 Matrices 193
9.1 Notations 193
9.2 Introduction 193
9.3 Skew Hadamard Matrices 193
9.4 Constructions for Skew Hadamard Matrices 195
9.5 Szekeres Difference Sets 200
9.6 Amicable Hadamard Matrices 204
9.7 Amicable Cores 207
9.8 Construction for Amicable Hadamard Matrices of Order 2t 208
9.9 Construction of Amicable Hadamard Matrices Using Cores 209
9.10 Symmetric Hadamard Matrices 211

10 Skew Hadamard Difference Sets 215
10.1 Notations 215
10.2 Skew Hadamard Difference Sets 215
10.3 The Construction by Planar Functions Over a Finite Field 215
10.4 The Construction by Using Index 2 Gauss Sums 218
10.5 The Construction by Using Normalized Relative Gauss Sums 226

11 Asymptotic Existence of Hadamard Matrices 233
11.1 Notations 233
11.2 Introduction 233
11.3 Seberry's Theorem 233
11.4 Craigen's Theorem 234
11.5 More Asymptotic Theorems 243
11.6 Skew Hadamard and Regular Hadamard 243

12 More on Maximal Determinant Matrices 245
12.1 Notations 245
12.2 E-Equivalence: The Smith Normal Form 245
12.3 E-Equivalence: The Number of Small Invariants 247
12.4 E-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
12.7 D-Optimal Matrices Embedded in Hadamard Matrices 254
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 Order n − 8 260
12.12 Embeddability of Hadamard Matrices of Order n − k 261
12.13 Growth Problem for Hadamard Matrices 265

A Hadamard Matrices 271
B List of sds from Cyclotomy 295
C Further Research Questions 301

References 303
Index 313