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Hadamard Matrices

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Up-to-date resource on Hadamard matricesHadamard Matrices: Constructions using Number Theory and Algebra provides students with a ... Weiterlesen
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Beschreibung

Up-to-date resource on Hadamard matricesHadamard 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 matricesThe 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.

Autorentext
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.

Klappentext
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: Constructions Using Number Theory and Algebra 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.

Zusammenfassung
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.

Inhalt
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 identical to 7 (mod 8) 219 10.4.3 The Case that p1 identical to 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 : lambda} 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

Produktinformationen

Titel: Hadamard Matrices
Untertitel: Constructions using Number Theory and Linear Algebra
Autor:
EAN: 9781119520245
ISBN: 978-1-119-52024-5
Format: Fester Einband
Herausgeber: John Wiley and Sons Ltd
Genre: Mathematik
Anzahl Seiten: 352
Gewicht: 990g
Größe: H212mm x B263mm x T26mm
Jahr: 2020
Untertitel: Englisch
Auflage: 1. Auflage