Up-to-date resource on Hadamard matrices
Hadamard Matrices: Constructions using Number Theory and Algebra provides students with a discussion of the basic definitions used for Hadamard Matrices as well as more advanced topics in the subject, including:
- Gauss sums, Jacobi sums and relative Gauss sums
- Cyclotomic numbers
- Plug-in matrices, arrays, sequences and M-structure
- Galois rings and Menon Hadamard differences sets
- Paley difference sets and Paley type partial difference sets
- Symmetric Hadamard matrices, skew Hadamard matrices and amicable Hadamard matrices
- A discussion of asymptotic existence of Hadamard matrices
- Maximal determinant matrices, embeddability of Hadamard matrices and growth problem for Hadamard matrices
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 Matrices combines 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.
Table of Contents
List of Tables xiii
List of Figures xv
Preface xvii
Acknowledgments xix
Acronyms xxi
Introduction xxiii
1 Basic Definitions 1
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
1.16 T-matrices, T-sequences, and Turyn Sequences 47
1.16.1 Turyn Sequences 48
2 Gauss Sums, Jacobi Sums, and Relative Gauss Sums 49
2.1 Notations 49
2.2 Gauss Sums 49
2.3 Jacobi Sums 51
2.3.1 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 for e = 2 62
2.4.2 Cyclotomic Numbers for e = 4 63
2.4.3 Cyclotomic Numbers for e = 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 Primep 72
2.6.2 Stickelberger’s Theorem 72
2.6.3 Prime Ideal Factorization of the Gauss Sum in Q(𝜁q-1) 73
2.6.4 Prime Ideal Factorization of the Gauss Sums in Q(𝜁m) 74
3 Plug-In Matrices 77
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 Goethals-Seidel 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 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
3.7.1 Whiteman’s 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-Into 115
4.1 Notations 115
4.2 Orthogonal Designs 115
4.2.1 Baumert-Hall Arrays and Welch Arrays 116
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
5.3.1 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,±B Matrices 127
5.3.4 Matrix Specifics 129
5.3.5 Counting Two Ways 129
5.3.6 For m Odd: 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 Yang’s Theorems on T-sequences 140
5.10.2 Multiplying by 2g + 1, g the Length of a Golay Sequence 142
5.10.3 Multiplying by 7 and 13 143
5.10.4 Koukouvinos and Kounias Number 144
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 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 Over GR(2n+1, s) 171
7.6.5 Menon Hadamard Difference Sets Over GR(2n+1, s) 172
7.6.6 Menon Hadamard Difference Sets Over GR(22, s) 173
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
8.3.1 Stanton-Sprott Difference Sets 179
8.3.2 Paley Hadamard Difference Sets Obtained from Relative Gauss Sums 180
8.3.3 Gordon-Mills-Welch 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 Matrices 193
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 Goethals-Seidel Type 196
9.4.2 An Adaption of Wallis-Whiteman 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 2t 208
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 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
10.3.1 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 that p1 ≡ 7 (mod 8) 219
10.4.3 The Case that p1 ≡ 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 Matrices 233
11.1 Notations 233
11.2 Introduction 233
11.2.1 de Launey’s Theorem 233
11.3 Seberry’s Theorem 233
11.4 Craigen’s 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 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
12.7.1 Embedding of D5 in H8 254
12.7.2 Embedding of D6 in H8 255
12.7.3 Embedding of D7 in H8 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 Order n - 8 260
12.12 Embeddability of Hadamard Matrices of Order n -k 261
12.12.1 Embeddability-Extendability of Hadamard Matrices 262
12.12.2 Available Determinant Spectrum and Verification 263
12.13 Growth Problem for Hadamard Matrices 265
A Hadamard Matrices 271
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 Craigen-Seberry-Zhang 274
A.1.13 de Launey 274
A.1.14 Seberry/Craigen Asymptotic Theorems 275
A.1.15 Yang’s 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 Cyclotomy 295
B.1 Introduction 295
B.2 List of n - {q; k1,…, kn ∶ 𝜆} sds 295
C Further Research Questions 301
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 Launey’s 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
