Hadamard Matrices

Preface xxv Acknowledgments xxvii Acronyms xxix Introduction iii 1 Basic Definitions 1 1.1 Notations 1 1.2 Finite Fields 5 1.2.1 A Residue Class Ring 5 1.2.2 Properties of Finite Fields 7 1.2.3 Traces and Norms 8 1.2.4 Characters of Finite Fields 12 1.3 Group Rings and Their Characters 16 1.4 Type 1 and Type 2 Matrices 17 1.5 Hadamard Matrices 25 1.5.1 Definition and Properties of an Hadamard Matrix 25 1.5.2 Kronecker Product and the Sylvester Hadamard Matrices 31 1.5.3 Inequivalence Classes 36 1.6 Paley Core Matrices 36 1.7 Amicable Hadamard Matrices 40 1.8 The Additive Property and Four Plug-in Matrices 46 1.8.1 Computer Construction 48 1.8.2 Skew Hadamard Matrices 48 1.8.3 Symmetric Hadamard Matrices 49 1.9 Difference Sets, SDS and Partial Difference Sets 51 1.9.1 Difference Sets 51 1.9.2 Supplementary Difference Sets 54 1.9.3 Partial Difference Sets 57 1.10 Sequences and Autocorrelation Function 59 1.10.1 Multiplication of NPAF Sequences 63 1.10.2 Golay Sequences 65 1.11 Excess 68 1.12 Balanced Incomplete Block Designs 70 1.13 Hadamard Matrices and SBIBDs 73 1.14 Cyclotomic Numbers 75 The case where e = 2 76 The case where e = 4 and f is even 77 The case where e = 4 and f is odd 78 The case where e = 8 and f is odd 79 The case where e = 8 and f is even 81 1.15 Orthogonal Designs and Weighing Matrices 83 1.16 T -matrices, T -sequences and Turyn Sequences 84 1.16.1 Turyn Sequences 86 2 Gauss Sums, Jacobi Sums and Relative Gauss Sums 89 2.1 Notations 89 2.2 Gauss Sums 89 2.3 Jacobi Sums 93 2.3.1 Congruence Relations 94 2.3.2 Jacobi Sums of Order 4 95 2.3.3 Jacobi Sums of Order 8 101 2.4 Cyclotomic Numbers and Jacobi Sums 106 2.4.1 Cyclotomic Numbers for e = 2 110 2.4.2 Cyclotomic Numbers for e = 4 111 2.4.3 Cyclotomic Numbers for e = 8 113 2.5 Relative Gauss Sums 120 2.6 Prime Ideal Factorization of Gauss Sums 126 2.6.1 Prime Ideal Factorization of a Prime p 126 2.6.2 Stickelberger's Theorem 126 2.6.3 Prime Ideal Factorization of the Gauss Sum in Q(zetaq.1) 128 2.6.4 Prime Ideal Factorization of the Gauss Sums in Q(zetam) 130 3 Plug-In Matrices 133 3.1 Notations 133 3.2 Williamson Type and Williamson Matrices 135 3.3 Plug-in Matrices 143 3.3.1 The Ito Array 144 3.3.2 Good Matrices : a Variation of Williamson Matrices 144 3.3.3 The Goethals-Seidel Array 146 3.3.4 Symmetric Hadamard Variation 146 3.4 Eight Plug-in Matrices 147 3.4.1 The Kharaghani Array 148 3.5 More T -sequences and T -matrices 149 3.6 Construction of T -matrices of Order 6m+ 1 153 3.7 Williamson Hadamard Matrices and Paley Type II Hadamard Matrices 159 3.7.1 Whiteman's Construction 159 3.7.2 Williamson Equation from Relative Gauss Sums 164 3.8 Hadamard Matrices of Generalized Quaternion Type 169 3.8.1 Definitions 169 3.8.2 Paley Core Type I Matrices 172 3.8.3 Hadamard Matrices of GQ Type and Relative Gauss Sums 172 3.9 Supplementary Difference Sets and Williamson Matrices 174 3.9.1 Supplementary Difference Sets from Cyclotomic Classes 174 3.9.2 Constructions of an Hadamard 4-sds 177 3.9.3 Construction from (q; x, y)-Partitions 183 3.10 Relative DS and Williamson-Type Matrices over Abelian Groups 191 3.11 Computer Construction of Williamson Matrices 196 4 Arrays: Matrices to Plug-Into 199 4.1 Notations 199 4.2 Orthogonal Designs 200 4.2.1 Baumert-Hall Arrays and Welch Arrays 201 4.3 Welch and Ono-Sawade-Yamamoto Arrays 210 4.4 Regular Representation of a Group and BHW(G) 212 5 Sequences 217 5.1 Notations 217 5.2 PAF and NPAF 219 5.3 Suitable Single Sequences 220 5.3.1 On Circulant Hadamard matrices for Orders >4 220 5.3.2 SBIBD Implications 221 5.3.3 From ±1 Matrices to ±A, ±B Matrices 222 5.3.4 Matrix Specifics 224 5.3.5 Counting Two Ways 225 5.3.6 For m Odd: Orthogonal Design Implications 227 5.3.7 The Case for Order 16 228 5.4 Suitable Pairs of NPAF Sequences: Golay Sequences 229 5.5 Current Results for Golay Pairs 229 5.6 Recent Results for Periodic Golay Pairs 232 5.7 More on Four Complementary Sequences 232 5.8 6-Turyn-type Sequences 237 5.9 Base Sequences 238 5.10 Yang-Sequences 240 5.10.1 On Yang's Theorems on T -sequences 247 5.10.2 Multiplying by 2g + 1, g the Length of a Golay Sequence 248 5.10.3 Multiplying by 7 and 13 249 5.10.4 Koukouvinos and Kounias Number 251 6 M-structures 257 6.1 Notations 257 6.2 The Strong Kronecker Product 258 6.3 Reducing the Powers of Two 261 6.4 Multiplication Theorems Using M-structures 265 6.5 Miyamoto's Theorem and Corollaries via M-structures 269 7 Menon Hadamard Difference Sets 283 7.1 Notations 283 7.2 Menon Hadamard Difference Sets and Exponent Bound 285 7.3 Menon Hadamard Difference Sets and Relative Hadamard Matrices 286 7.4 The Constructions from Cyclotomy 288 7.5 The Constructions Using Projective Sets 294 7.5.1 Graphical Hadamard Matrices 301 7.6 The Construction Based on Galois Rings 303 7.6.1 Galois Rings 303 7.6.2 Additive Characters of Galois Rings 304 7.6.3 A New Operation 306 7.6.4 Gauss Sums over GR(2^n+1, s) 306 7.6.5 Menon Hadamard Difference Sets over GR(2^n+1, s) 307 7.6.6 Menon Hadamard Difference Sets over GR(2², s) 309 8 Paley Hadamard Difference Sets and Partial Difference Sets 311 8.1 Notations 311 8.2 Paley Core Matrices and Gauss Sums 313 8.3 Paley Hadamard Difference Sets 317 8.3.1 Stanton-Sprott Difference Sets 318 8.3.2 Paley Hadamard Difference Sets and Relative Gauss Sums 320 8.3.3 Gordon-Mills-Welch Extension 323 8.4 Paley Type Partial Difference Set 324 8.5 Paley Type PDS from EBS 326 8.6 Constructing Paley Hadamard Difference Sets 341 9 Skew Hadamard, Amicable and Symmetric Matrices 345 9.1 Notations 345 9.2 Introduction 346 9.3 Skew Hadamard Matrices 347 9.3.1 Summary of Skew Hadamard Orders 347 9.4 Constructions for Skew Hadamard Matrices 349 9.4.1 The Goethals-Seidel Type 352 9.4.2 An Adaption of Wallis-Whiteman Array 353 9.5 Szekeres Difference Sets 359 9.5.1 The Construction by Cyclotomic Numbers 362 9.6 Amicable Hadamard Matrices 365 9.7 Amicable Cores 370 9.8 Construction for Amicable Hadamard Matrices of Order 2¯t 372 9.9 Construction of Amicable Hadamard Matrices Using Cores 374 9.10 Symmetric Hadamard Matrices 378 9.10.1 Symmetric Hadamard Matrices via Computer Construction 379 9.10.2 Luchshie Matrices Known Results 380 10 Skew Hadamard Difference Sets 383 10.1 Notations 383 10.2 Skew Hadamard Difference Sets 383 10.3 The Construction by Planar Functions over a Finite Field 385 10.3.1 Planar Functions and Dickson Polynomials 385 10.4 The Construction by Using Index 2 Gauss Sums 388 10.4.1 Index 2 Gauss Sums 388 10.4.2 The Case that p1 identical to 7 (mod 8) 390 10.4.3 The Case that p1 identical to 3 (mod 8) 394 Case 1 396 Case 2 396 Case 3 396 Case 4 397 Case 5 397 10.5 The Construction by using Normalized Relative Gauss Sums 402 10.5.1 More on Ideal Factorization of the Gauss Sum 402 10.5.2 Determination of Normalized Relative Gauss Sums 403 10.5.3 A Family of Skew Hadamard Difference Sets 407 11 Asymptotic Existence of Hadamard Matrices 411 11.1 Notations 411 11.2 Introduction 412 11.2.1 de Launey's Theorem 412 11.3 Seberry's Theorem 412 11.4 Craigen's Theorem 413 11.4.1 Signed Groups and Their Representations 414 11.4.2 A Construction for Signed Group Hadamard Matrices 417 11.4.3 A Construction for Hadamard Matrices 421 11.4.4 Comments on Orthogonal Matrices Over Signed Groups 424 11.4.5 Some Calculations 426 11.5 More Asymptotic Theorems 431 11.6 Skew Hadamard and Regular Hadamard 431 12 More on Maximal Determinant Matrices 433 12.1 Notations 433 12.2 E-Equivalence: The Smith Normal Form 434 12.3 E-Equivalence: The Number of Small Invariants 438 12.4 E-Equivalence: Skew Hadamard and Symmetric Conference Matrices 443 12.5 Smith Normal Form for Powers of 2 446 12.6 Matrices with Elements (1,.1) and Maximal Determinant 448 12.7 D-optimal Matrices Embedded in Hadamard Matrices 449 12.7.1 Embedding of D5 in H8 451 12.7.2 Embedding of D6 in H8 451 12.7.3 Embedding of D7 in H8 452 12.7.4 Other Embeddings 453 12.8 Embedding of Hadamard Matrices within Hadamard Matrices 455 12.9 Embedding Properties via Minors 456 12.10Embeddability of Hadamard Matrices 459 12.11Embeddability of Hadamard Matrices of Order n . 8 461 12.12Embeddability of Hadamard Matrices of Order n . k 464 12.12.1Embeddability--Extendability of Hadamard Matrices 467 12.12.2Available Determinant Spectrum and Verification 468 12.13Growth Problem for Hadamard Matrices 472 The Pivot Structure of H8 474 The Pivot Structure of H12 475 The Pivot Structure of H16 476 A Hadamard Matrices 479 A.1 Hadamard Matrices 479 A.1.1 Amicable Hadamard Matrices 479 A.1.2 Skew Hadamard Matrices 480 A.1.3 Spence Hadamard Matrices 480 A.1.4 Conference Matrices Give Symmetric Hadamard Matrices 481 A.1.5 Hadamard Matrices from Williamson Matrices 481 A.1.6 OD Hadamard Matrices 481 A.1.7 Yamada Hadamard Matrices 482 A.1.8 Miyamoto Hadamard Matrices 482 A.1.9 Koukouvinos and Kounias 483 A.1.10 Yang Numbers 483 A.1.11 Agaian Multiplication 483 A.1.12 Craigen-Seberry-Zhang 483 A.1.13 de Launey 483 A.1.14 Seberry/Craigen Asymptotic Theorems 484 A.1.15 Yang's Theorems and okovi¿c updates 484 A.1.16 Computation by okovi¿c 484 A.2 Index of Williamson Matrices 486 A.3 Tables of Hadamard Matrices 499 B List of sds from Cyclotomy 515 B.1 Introduction 515 B.2 List of n . {q; k1, , kn: lambda} sds. 515 C Further Research Questions 525 C.1 Research Questions for Future Investigation 525 C.1.1 Matrices 525 C.1.2 Base Sequences 525 C.1.3 Partial Difference Sets 526 C.1.4 de Launey's Four Questions 526 C.1.5 Embedding Sub-matrices 527 C.1.6 Pivot Structures 527 C.1.7 Trimming and Bordering 527 C.1.8 Arrays 527 Index 553