Hadamard Matrices and Their Applications
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1168
8.8. Agaian
Hadamard Matrices and Their Applications
Springer-Verlag Berlin Heidelberg New York Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: USSR Adviser: L. D. Faddeev, Leningrad
1168
8.8. Agaian
Hadamard Matrices and Their Applications
Springer-Verlag Berlin Heidelberg New York Tokyo
Author S.S. Agaian Computer Center of the Academy of Sciences Sevak str. 1, Erevan 44, USSR Consulting Editor D. Yu. Grigorev Leningrad Branch of the Steklov Mathematical Institute Fontanka 27, 191011 Leningrad, D-11, USSR
Mathematics Subject Classification (1980): 05XX; 05BXX ISBN 3-540-16056-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16056-6 Springer-Verlag New York Heidelberg Berlin Tokyo
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
2146/3140·543210
CONTENTS
Introduction •••••••••••.••••••••••••••.•.•••••••••••••••••••••••
1
§ 1. Basic definitions, notations and auxiliary results •••
5
Chapter 1. CONSTRUCTION OF CLASSIC HADAMARD MATRICES ••••.•••••••
11
§ 2. Methods of construction for Hadamard matrices ••••••••
11
§ 3. Some problems of construction for Hadamard matrices ••
49
§ 4. New method for Hadamard matrices construction ••••••••
78
Chapter 2. CONSTRUCTION OF GENERALIZED HADAMARD MATRICES ••.••••• 103 § 5. Generalized Hadamard matrices
103
§ 6. Construction of high-dimensional Hadamard matrices ••• 114
Chapter 3. APPLICATION
OF HADAMARD MATRICES •.•.•.•••••••••.•••• 134
§ 7. Hadamard matrices and problems of information Theory • 134 § 8. Hadamard matrices and design theory •...•••••••••.•••• 166 § 9. Other applications of Hadamard matrices .•••••...••••• 1 71
Appendix 1. UNANSWERED PROBLEMS •••••••••.••••••••••••••••••••••• 178 Appendix 2. TABLES OF BLOCK-CIRCULANT, BLOCK-SYMMETRIC (PLANE AND HIGH-DIMENSIONAL) HADAMARD MATRICES OF ORDER (4n) .180 References ••••••••••••••••••••••••••••..••.•.••.•••••••••••.•••• 192 Subject Index ..•.••••.....•••...•.•••...•.......••.•••.••...•••. 216
Introduction
The importance of
ortho0onal matrices in modern discrete mathe-
matics and. its applications is well known; for many applied problems (for example, for construction of discrete equipments realizing fast or orthogonal transformations) one needs consideration of integer orthogonal matrices and in particular, orthogonal matrices with the elements -1 and +1. Square orthogonal matrices with the elements -1 and +1 are called Hadamard matrices. Investigations of Hadamard matrices were connected initially with a linear algebra problems (for ex
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