Hadamard Matrices - Constructions Using Number Theory and Linear Algebra

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.

Inhaltsverzeichnis

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(zeta q-1) 73
 
2.6.4 Prime Ideal Factorization of the Gauss Sums in Q(zeta m) 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)-Partit

Über den Autor / die Autorin

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.

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.

Produktdetails

• Autoren: J Seberry, Jennife Seberry, Jennifer Seberry, Jennifer Yamada Seberry, Seberry Jennifer, Mieko Yamada, Yamada Mieko
• Verlag: Wiley, John and Sons Ltd
• Sprache: Englisch
• Produktform: Fester Einband
• Erschienen: 31.10.2020
• EAN: 9781119520245
• ISBN: 978-1-119-52024-5
• Seiten: 352
• Themen: Naturwissenschaften, Medizin, Informatik, Technik > Mathematik > Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik; Mathematik, Zahlentheorie, Signalverarbeitung, MATHEMATICS / Number Theory, Lineare Algebra, Mathematics, Linear Algebra, Number Theory, Signal Processing, Electrical & Electronics Engineering, Elektrotechnik u. Elektronik, Hadamard-Matrix