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Algebraic Design Theory Mathematical Surveys and Monographs Volume 175 Algebraic Design Theory Warwick de Launey Dane Flannery American Mathematical Society http://dx.doi.org/10.1090/surv/175 Mathematical Surveys and Monographs Volume 175 Algebraic Design Theory Warwick de Launey Dane Flannery American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Jordan S. Ellenberg Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 05-02, 05Bxx, 05E18, 16B99, 20Dxx; Secondary 05-04, 15A24, 16S99, 20B20, 20J06. For additional information and updates on this book, visit www.ams.org/bookpages/surv-175 Library of Congress Cataloging-in-Publication Data De Launey, Warwick, 1958– Algebraic design theory / Warwick De Launey, Dane Flannery. p. cm. — (Mathematical surveys and monographs ; v. 175) Includes bibliographical references and index. ISBN 978-0-8218-4496-0 (alk. paper) 1. Combinatorial designs and configurations. I. Flannery, D. L. (Dane Laurence), 1965– II. Title QA166.25.D43 2011 511.6–dc23 2011014837 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 To Scott Godfrey MD, Richard Lam MD, and Mark Scholz MD — Warwick de Launey To my parents, Lois and Ivan — Dane Flannery Contents Preface ix Chapter 1. Overview 1 1.1. What is a combinatorial design? 1 1.2. What is Algebraic Design Theory? 1 1.3. What is in this book? 2 Chapter 2. Many Kinds of Pairwise Combinatorial Designs 7 2.1. Orthogonality sets 7 2.2. Symmetric balanced incomplete block designs 9 2.3. Hadamard matrices 11 2.4. Weighing matrices 12 2.5. Balanced weighing matrices 13 2.6. Orthogonal designs 14 2.7. Complex Hadamard matrices 16 2.8. Complex generalized Hadamard matrices 18 2.9. Complex generalized weighing matrices 19 2.10. Generalized Hadamard matrices over groups 19 2.11. Balanced generalized weighing matrices 22 2.12. Generalized weighing matrices 23 2.13. Summary 24 Chapter 3. A Primer for Algebraic Design Theory 27 3.1. Groups 27 3.2. Monoids 34 3.3. Group actions 35 3.4. Rings 39 3.5. Matrices 41 3.6. Linear and related groups 42 3.7. Representations 44 Chapter 4. Orthogonality 49 4.1. How many rows can be pairwise Λ-orthogonal? 49 4.2. Non-trivial orthogonality sets 50 4.3. A big picture 51 4.4. Equivalence 54 4.5. Matrices, arrays, and designs 60 Chapter 5. Modeling Λ-Equivalence 63 5.1. A first look at the automorphism group 63 5.2. Ambient rings with a model for Λ-equivalence 65 v vi CONTENTS 5.3. Ambient rings for the familiar orthogonality sets 68 Chapter 6. The Grammian 71 6.1. Orthogonality as a Grammian property 71 6.2. Non-degeneracy 72 6.3. Gram completions and composition of orthogonality sets 73 6.4. The Gram Property and Λ-equivalence 74 Chapter 7. Transposability 77 7.1. The main problems 77 7.2. A functional approach to self-duality 78 7.3. Conjugate equivalence operations 80 7.4. A matrix algebra approach to transposability and self-duality 80 7.5. A different kind of transposable orthogonality set 82 Chapter 8. New Designs from Old 85 8.1. Composition 85 8.2. Transference 93 Chapter 9. Automorphism Groups 99 9.1. Automorphism groups of pairwise combinatorial designs 99 9.2. A class of generalized Hadamard matrices 100 9.3. A bound on the size of the automorphism group 103 9.4. Permutation automorphism groups 105 9.5. Automorphism groups of orthogonal designs 106 9.6. Expanded designs 108 9.7. Computing automorphism groups 112 9.8. The associated design 114 9.9. Associated designs and group divisible designs 116 9.10. An isomorphism for weighing matrices 117 Chapter 10. Group Development and Regular Actions on Arrays 119 10.1. Matrix preliminaries 119 10.2. Group-developed arrays 119 10.3. Regular embeddings 121 10.4. Difference sets and relative difference sets 124 10.5. Group ring equations and associates 127 10.6. Finding all associates of an array 129 10.7. An algorithm for solving Problems 10.2.3 and 10.2.4 131 10.8. Composition via associates 132 Chapter 11. Origins of Cocyclic Development 135 11.1. First derivation 135 11.2. Second derivation 140 11.3. Cocycles for cyclic groups 142 Chapter 12. Group Extensions and Cocycles 145 12.1. Central extensions 145 12.2. Cocycles for product groups 150 12.3. Polycyclic presentations 151 12.4. Cocycles from collection in polycyclic groups 153 CONTENTS vii 12.5. Monomial representations and cocycles 157 Chapter 13. Cocyclic Pairwise Combinatorial Designs 161 13.1. The main definitions 161 13.2. Ambient rings with a central group 162 13.3. Some big problems 164 13.4. Central extensions of a design 164 13.5. Approaches to cocyclic designs 165 Chapter 14. Centrally Regular Actions 167 14.1. Cocyclic forms 167 14.2. A lesser expanded design 167 14.3. A pair of lifting homomorphisms 168 14.4. The lift 169 14.5. Translation 170 14.6. Centrally regular embeddings 171 14.7. Finding cocyclic forms 173 14.8. All the cocycles of a design 176 Chapter 15. Cocyclic Associates 177 15.1. Definition of cocyclic associates 177 15.2. The group ring equation for cocyclic associates 178 15.3. The familiar designs 180 15.4. Cocyclic designs and relative difference sets 181 15.5. Normal p-complements 182 15.6. Existence conditions for cocyclic Hadamard matrices 183 15.7. Cyclotomic rings and circulant complex Hadamard matrices 185 15.8. Composition of cocyclic associates 190 Chapter 16. Special Classes of Cocyclic Designs 195 16.1. Cocyclic Hadamard matrices 195 16.2. Cocyclic weighing matrices 197 16.3. Cocyclic orthogonal designs 198 16.4. A cocyclic substitution scheme 200 16.5. Cocyclic complex Hadamard matrices 201 Chapter 17. The Paley Matrices 203 17.1. Actions of 2-dimensional linear and semilinear groups 203 17.2. The Paley matrices and their automorphism groups 205 17.3. The regular actions 209 Chapter 18. A Large Family of Cocyclic Hadamard Matrices 215 18.1. On the orders covered 215 18.2. A construction for prime powers congruent to 3 (mod 4) 216 18.3. A construction for prime powers congruent to 1 (mod 4) 218 18.4. Plug-in matrices 220 18.5. Proof of the main theorem and a generalization 221 Chapter 19. Substitution Schemes for Cocyclic Hadamard Matrices 223 19.1. General substitution schemes 224 19.2. Number-theoretic constraints 226 viii CONTENTS 19.3. Further results for group-developed plug-in matrices 227 19.4. Inverting action 228 19.5. Trivial action 230 19.6. Complementary pairs and the Cocyclic Hadamard Conjecture 232 19.7. Existence of group-developed complementary pairs 233 Chapter 20. Calculating Cocyclic Development Rules 239 20.1. Introduction to development tables 239 20.2. Development tables for abelian groups 240 20.3. Development tables revisited 241 20.4. Group cohomology 242 20.5. Constructing a free table 243 20.6. Group homology 244 20.7. Presentations and the Schur multiplier 246 20.8. Constructing a torsion table 249 20.9. Listing the elements of the second cohomology group 253 20.10. Another look at the Cocyclic Hadamard Conjecture 255 Chapter 21. Cocyclic Hadamard Matrices Indexed by Elementary Abelian Groups 257 21.1. Motivation: indexing groups for the Sylvester matrices 257 21.2. The extension problem 258 21.3. Pure Hadamard collection cocycles 261 21.4. Bilinearity and Hadamard cocycles 262 21.5. Solution of the Hadamard cocycle problem 263 Chapter 22. Cocyclic Concordant Systems of Orthogonal Designs 267 22.1. Existence and uniqueness of cocyclic systems of OD(n;1k) 267 22.2. A reduction 268 22.3. Solution of the reduced problem 269 22.4. Proof of Theorem 22.1.1 270 22.5. Removing the zeros 271 22.6. Examples 272 Chapter 23. Asymptotic Existence of Cocyclic Hadamard Matrices 279 23.1. Complex sequences with zero aperiodic autocorrelation 279 23.2. Sets of Hermitian and skew-Hermitian circulant matrices 281 23.3. Sets of cocyclic signed permutation matrices 282 23.4. Existence of cocyclic complex Hadamard matrices 283 23.5. Concluding remarks 284 Bibliography 287 Index 295 Preface Over the past several decades, algebra has become increasingly important in combinatorial design theory. The flow of ideas has for the most part been from algebra to design theory. Moreover, despite our successes, fundamental algebraic questions in design theory remain open. It seems that new or more sophisticated ideas and techniques will be needed to make progress on these questions. In the meantime, design theory is a fertile source of problems that are ideal for spurring the development of algorithms in the active field of computational algebra.
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