Mathematical Surveys

and Monographs

Volume 175

American Mathematical Society

Algebraic Design Theory

Warwick de LauneyDane Flannery

Mathematical Surveys

and Monographs

Volume 175

Algebraic Design Theory

Warwick de Launey

Dane Flannery

American Mathematical SocietyProvidence, Rhode Island

http://dx.doi.org/10.1090/surv/175

EDITORIAL COMMITTEE

Ralph L. Cohen, ChairJordan S. Ellenberg

Michael A. SingerBenjamin Sudakov

Michael I. 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, visitwww.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 2011511′.6–dc23 2011014837

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made bye-mail to reprint-permission@ams.org.

c© 2011 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rightsexcept 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 guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 11

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 11.1. What is a combinatorial design? 11.2. What is Algebraic Design Theory? 11.3. What is in this book? 2

Chapter 2. Many Kinds of Pairwise Combinatorial Designs 72.1. Orthogonality sets 72.2. Symmetric balanced incomplete block designs 92.3. Hadamard matrices 112.4. Weighing matrices 122.5. Balanced weighing matrices 132.6. Orthogonal designs 142.7. Complex Hadamard matrices 162.8. Complex generalized Hadamard matrices 182.9. Complex generalized weighing matrices 192.10. Generalized Hadamard matrices over groups 192.11. Balanced generalized weighing matrices 222.12. Generalized weighing matrices 232.13. Summary 24

Chapter 3. A Primer for Algebraic Design Theory 273.1. Groups 273.2. Monoids 343.3. Group actions 353.4. Rings 393.5. Matrices 413.6. Linear and related groups 423.7. Representations 44

Chapter 4. Orthogonality 494.1. How many rows can be pairwise Λ-orthogonal? 494.2. Non-trivial orthogonality sets 504.3. A big picture 514.4. Equivalence 544.5. Matrices, arrays, and designs 60

Chapter 5. Modeling Λ-Equivalence 635.1. A first look at the automorphism group 635.2. 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 716.1. Orthogonality as a Grammian property 716.2. Non-degeneracy 726.3. Gram completions and composition of orthogonality sets 736.4. The Gram Property and Λ-equivalence 74

Chapter 7. Transposability 777.1. The main problems 777.2. A functional approach to self-duality 787.3. Conjugate equivalence operations 807.4. A matrix algebra approach to transposability and self-duality 807.5. A different kind of transposable orthogonality set 82

Chapter 8. New Designs from Old 858.1. Composition 858.2. Transference 93

Chapter 9. Automorphism Groups 999.1. Automorphism groups of pairwise combinatorial designs 999.2. A class of generalized Hadamard matrices 1009.3. A bound on the size of the automorphism group 1039.4. Permutation automorphism groups 1059.5. Automorphism groups of orthogonal designs 1069.6. Expanded designs 1089.7. Computing automorphism groups 1129.8. The associated design 1149.9. Associated designs and group divisible designs 1169.10. An isomorphism for weighing matrices 117

Chapter 10. Group Development and Regular Actions on Arrays 11910.1. Matrix preliminaries 11910.2. Group-developed arrays 11910.3. Regular embeddings 12110.4. Difference sets and relative difference sets 12410.5. Group ring equations and associates 12710.6. Finding all associates of an array 12910.7. An algorithm for solving Problems 10.2.3 and 10.2.4 13110.8. Composition via associates 132

Chapter 11. Origins of Cocyclic Development 13511.1. First derivation 13511.2. Second derivation 14011.3. Cocycles for cyclic groups 142

Chapter 12. Group Extensions and Cocycles 14512.1. Central extensions 14512.2. Cocycles for product groups 15012.3. Polycyclic presentations 15112.4. Cocycles from collection in polycyclic groups 153

CONTENTS vii

12.5. Monomial representations and cocycles 157

Chapter 13. Cocyclic Pairwise Combinatorial Designs 16113.1. The main definitions 16113.2. Ambient rings with a central group 16213.3. Some big problems 16413.4. Central extensions of a design 16413.5. Approaches to cocyclic designs 165

Chapter 14. Centrally Regular Actions 16714.1. Cocyclic forms 16714.2. A lesser expanded design 16714.3. A pair of lifting homomorphisms 16814.4. The lift 16914.5. Translation 17014.6. Centrally regular embeddings 17114.7. Finding cocyclic forms 17314.8. All the cocycles of a design 176

Chapter 15. Cocyclic Associates 17715.1. Definition of cocyclic associates 17715.2. The group ring equation for cocyclic associates 17815.3. The familiar designs 18015.4. Cocyclic designs and relative difference sets 18115.5. Normal p-complements 18215.6. Existence conditions for cocyclic Hadamard matrices 18315.7. Cyclotomic rings and circulant complex Hadamard matrices 18515.8. Composition of cocyclic associates 190

Chapter 16. Special Classes of Cocyclic Designs 19516.1. Cocyclic Hadamard matrices 19516.2. Cocyclic weighing matrices 19716.3. Cocyclic orthogonal designs 19816.4. A cocyclic substitution scheme 20016.5. Cocyclic complex Hadamard matrices 201

Chapter 17. The Paley Matrices 20317.1. Actions of 2-dimensional linear and semilinear groups 20317.2. The Paley matrices and their automorphism groups 20517.3. The regular actions 209

Chapter 18. A Large Family of Cocyclic Hadamard Matrices 21518.1. On the orders covered 21518.2. A construction for prime powers congruent to 3 (mod 4) 21618.3. A construction for prime powers congruent to 1 (mod 4) 21818.4. Plug-in matrices 22018.5. Proof of the main theorem and a generalization 221

Chapter 19. Substitution Schemes for Cocyclic Hadamard Matrices 22319.1. General substitution schemes 22419.2. Number-theoretic constraints 226

viii CONTENTS

19.3. Further results for group-developed plug-in matrices 22719.4. Inverting action 22819.5. Trivial action 23019.6. Complementary pairs and the Cocyclic Hadamard Conjecture 23219.7. Existence of group-developed complementary pairs 233

Chapter 20. Calculating Cocyclic Development Rules 23920.1. Introduction to development tables 23920.2. Development tables for abelian groups 24020.3. Development tables revisited 24120.4. Group cohomology 24220.5. Constructing a free table 24320.6. Group homology 24420.7. Presentations and the Schur multiplier 24620.8. Constructing a torsion table 24920.9. Listing the elements of the second cohomology group 25320.10. Another look at the Cocyclic Hadamard Conjecture 255

Chapter 21. Cocyclic Hadamard Matrices Indexed by Elementary AbelianGroups 257

21.1. Motivation: indexing groups for the Sylvester matrices 25721.2. The extension problem 25821.3. Pure Hadamard collection cocycles 26121.4. Bilinearity and Hadamard cocycles 26221.5. Solution of the Hadamard cocycle problem 263

Chapter 22. Cocyclic Concordant Systems of Orthogonal Designs 26722.1. Existence and uniqueness of cocyclic systems of OD(n; 1k) 26722.2. A reduction 26822.3. Solution of the reduced problem 26922.4. Proof of Theorem 22.1.1 27022.5. Removing the zeros 27122.6. Examples 272

Chapter 23. Asymptotic Existence of Cocyclic Hadamard Matrices 27923.1. Complex sequences with zero aperiodic autocorrelation 27923.2. Sets of Hermitian and skew-Hermitian circulant matrices 28123.3. Sets of cocyclic signed permutation matrices 28223.4. Existence of cocyclic complex Hadamard matrices 28323.5. Concluding remarks 284

Bibliography 287

Index 295

Preface

Over the past several decades, algebra has become increasingly important incombinatorial design theory. The flow of ideas has for the most part been fromalgebra to design theory. Moreover, despite our successes, fundamental algebraicquestions in design theory remain open. It seems that new or more sophisticatedideas and techniques will be needed to make progress on these questions. In themeantime, design theory is a fertile source of problems that are ideal for spurringthe development of algorithms in the active field of computational algebra.

We hope that this book will encourage the investigation, by researchers at alllevels, of the algebraic questions posed by design theory. To this end, we providea large selection of the algebraic objects and applications to be found in designtheory. We also isolate a small number of problems that we think are important.

This book is a technical work that takes an unusually abstract approach. Whilethe approach is non-standard, it offers uniformity and enables us to highlight theprincipal themes in such a way that they can be studied for their own sake, ratherthan as a means to an end in special cases.

Everything begins with the following notion of orthogonality. Fix an integerb > 1, and a non-empty set A (an ‘alphabet’) excluding zero. Let Λ be a set (an‘orthogonality set’) of 2 × b arrays whose non-zero entries come from A. Much ofdesign theory is concerned with instances of the question

When does there exist a v × b array D such that every 2 × bsubarray of D is in Λ?

If D exists, then we say that its rows are pairwise Λ-orthogonal. Since essentiallycombinatorial constraints are being placed on pairs of distinct rows, and becauseof antecedents in the design of experiments, we call D a pairwise combinatorialdesign, or PCD(v,Λ) for short. Chapter 2 describes families of widely-studied pair-wise combinatorial designs. These designs are of interest in diverse fields includingelectrical engineering, statistical analysis, and finite geometry.

This book develops a theory of square pairwise combinatorial designs, i.e., thosewith v = b. For such designs we use the abbreviated notation PCD(Λ). Each of theprincipal design-theoretic themes finds expression. The ‘ambient rings’ introducedin Chapter 5 allow the free interplay of these themes: orthogonality, equivalence,transposability, composition, transference, the proliferation of inequivalent designs,the automorphism group, and links to group ring (norm) equations.

We pay particular attention to designs that possess a type of regular groupaction. The acting group has a certain central subgroup Z, and the corresponding2-cocycles with coefficients in Z have a significant influence on properties of thedesign. Such a design is said to be cocyclic. This book contains a general theory for

ix

x PREFACE

cocyclic pairwise combinatorial designs, plus many case studies. Along the way, weencounter numerous classical designs and other well-known mathematical objects.

This is a book of ideas. It is our opinion that design theory is still—even now—in its infancy. Thus, at this stage, ideas are more valuable than a compendium ofour present state of knowledge (which will keep growing rapidly beyond the confinesof a single volume). We have aimed to stimulate a creative reader rather than tobe encyclopedic.

With respect to cocyclic designs, the chief omissions from our book are NoboruIto’s work on Hadamard groups; and work by Kathy Horadam, her colleagues, andher students.

Our book covers some of Ito’s results, but from a different perspective. Startingin the 1980s, Ito produced a sequence of papers identifying regular group actionson the expanded design of a Hadamard matrix. We are content to refer the readerto those papers.

The first author, together with Horadam, founded the theory of cocyclic designsin the early 1990s. Horadam and her school have since published many resultsfocusing on Hadamard, complex Hadamard, and generalized Hadamard matrices.That material is covered in Horadam’s engaging book [87]. There one will findtopics such as shift equivalence of cocycles, equivalence classes of relative differencesets, and the connection between generalized Hadamard matrices and presemifields,that are not in this book.

We have tried to make the book as accessible as possible; we especially hopethat our treatment of the new ideas is welcoming and open-ended. Proofs are givenfor nearly all results outside of the ‘algebraic primer’ chapter and the chapter onPaley matrices. The book also contains a wealth of examples and case studies whichshould persuade the reader that the concepts involved are worthy of pursuit.

Acknowledgments. We are indebted to K. T. Arasu, Robert Craigen, KathyHoradam, Hadi Kharaghani, S. L. Ma, Michael J. Smith, and Richard M. Stafford,whose collaborations with the first author form the basis of several chapters andsections.

We received useful advice and feedback from Joe Buhler, Alla Detinko, JohnDillon, Al Hales, Kathy Horadam, Bill Kantor, Padraig O Cathain, Dick Stafford,Tobias Rossmann, and Jennifer Seberry. We are grateful to everyone for their help.

Many thanks are due as well to Sergei Gelfand and Christine Thivierge of theAmerican Mathematical Society, who guided us toward publication.

Finally, we thank Science Foundation Ireland for financial assistance from theResearch Frontiers Programme and Mathematics Initiative 2007 (grants 08/RFP/MTH1331 and 07/MI/007).

On November 8, 2010, Warwick de Launey passed away after a long illness.This book represents Warwick’s vision for Design Theory, gained from his years

of experience and achievement in the subject. It was my privilege to share in thestruggle to bring this vision to a wider audience.

The support of Warwick’s wife, Ione Rummery, was constant throughout ourwriting of the book, and is deeply appreciated.

Warwick has dedicated the book to his doctors. Their care gave him the timehe needed to complete his vision.

Dane FlanneryMarch 27, 2011

Bibliography

1. V. Alvarez, J. A. Armario, M. D. Frau, and P. Real, The homological reduction method forcomputing cocyclic Hadamard matrices, J. Symbolic Comput. 44 (2009), no. 5, 558–570.

2. K. T. Arasu and W. de Launey, Two-dimensional perfect quaternary arrays, IEEE Trans.Inform. Theory 47 (2001), no. 4, 1482–1493.

3. K. T. Arasu, W. de Launey, and S. L. Ma, On circulant complex Hadamard matrices, Des.Codes Cryptogr. 25 (2002), no. 2, 123–142.

4. K. T. Arasu and Q. Xiang, On the existence of periodic complementary binary sequences,Des. Codes Cryptogr. 2 (1992), no. 3, 257–262.

5. E. F. Assmus, Jr. and C. J. Salwach, The (16, 6, 2) designs, Internat. J. Math. Math. Sci. 2(1979), no. 2, 261–281.

6. R. D. Baker, An elliptic semiplane, J. Combin. Theory Ser. A 25 (1978), no. 2, 193–195.7. A. Baliga and K. J. Horadam, Cocyclic Hadamard matrices over Zt × Z2

2 , Australas. J.Combin. 11 (1995), 123–134.

8. L. D. Baumert, Cyclic difference sets, Lecture Notes in Math., vol. 182, Springer-Verlag,

Berlin, 1971.9. G. Berman, Weighing matrices and group divisible designs determined by EG(t, pr), p > 2,

Utilitas Math. 12 (1977), 183–191.10. , Families of generalized weighing matrices, Canad. J. Math. 30 (1978), no. 5, 1016–

1028.11. T. Beth, D. Jungnickel, and H. Lenz, Design theory. Vol. I, Encyclopedia of Mathematics

and its Applications, vol. 69, Cambridge University Press, Cambridge, 1999.12. F. R. Beyl and J. Tappe, Group extensions, representations, and the Schur multiplicator,

Lecture Notes in Math., Springer-Verlag, Berlin, 1982.13. R. C. Bose, On the construction of balanced incomplete block designs, Ann. Eugenics 9

(1939), 353–399.14. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language,

J. Symbolic Comput. 24 (1997), no. 3-4, 235–265.15. B. W. Brock, Hermitian congruence and the existence and completion of generalized

Hadamard matrices, J. Combin. Theory Ser. A 49 (1988), no. 2, 233–261.16. A. T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894–898.17. , Relations among generalized Hadamard matrices, relative difference sets, and max-

imal length linear recurring sequences, Canad. J. Math. 15 (1963), 42–48.18. G. Cohen, D. Rubie, J. Seberry, C. Koukouvinos, S. Kounias, and M. Yamada, A survey

of base sequences, disjoint complementary sequences and OD(4t; t, t, t, t), J. Combin. Math.Combin. Comput. 5 (1989), 69–103.

19. C. J. Colbourn and W. de Launey, Difference matrices, The CRC handbook of combinatorialdesigns, CRC Press, Boca Raton, 1996, pp. 287–296.

20. R. Compton, R. Craigen, and W. de Launey, Unreal BH(n, 6)s and Hadamard matrices,preprint.

21. R. Craigen, Signed groups, sequences, and the asymptotic existence of Hadamard matrices,J. Combin. Theory Ser. A 71 (1995), no. 2, 241–254.

22. R. Craigen and W. de Launey, Generalized Hadamard matrices whose transposes are notgeneralized Hadamard matrices, J. Combin. Des. 17 (2009), no. 6, 456–458.

23. R. Craigen, W. H. Holzmann, and H. Kharaghani, On the asymptotic existence of complexHadamard matrices, J. Combin. Des. 5 (1997), no. 5, 319–327.

24. R. Craigen and H. Kharaghani, On the nonexistence of Hermitian circulant complexHadamard matrices, Australas. J. Combin. 7 (1993), 225–227.

287

288 BIBLIOGRAPHY

25. , A combined approach to the construction of Hadamard matrices, Australas. J. Com-bin. 13 (1996), 89–107.

26. , Weaving Hadamard matrices with maximum excess and classes with small excess,J. Combin. Des. 12 (2004), no. 4, 233–255.

27. R. Craigen, J. Seberry, and X. M. Zhang, Product of four Hadamard matrices, J. Combin.Theory Ser. A 59 (1992), no. 2, 318–320.

28. T. Czerwinski, On finite projective planes with a single (P, l) transitivity, J. Combin. Theory

Ser. A 48 (1988), no. 1, 136–138.29. J. E. Dawson, A construction for generalized Hadamard matrices GH(4q,EA(q)), J. Statist.

Plann. Inference 11 (1985), no. 1, 103–110.30. D. de Caen, D. A. Gregory, and D. Pritikin, Minimum biclique partitions of the complete

multigraph and related designs, Graphs, matrices, and designs, Lecture Notes in Pure andAppl. Math., vol. 139, Dekker, New York, 1993, pp. 93–119.

31. D. de Caen, R. Mathon, and G. E. Moorhouse, A family of antipodal distance-regular graphsrelated to the classical Preparata codes, J. Algebraic Combin. 4 (1995), no. 4, 317–327.

32. W. de Launey, Generalised Hadamard matrices whose rows and columns form a group,Combinatorial mathematics, X (Adelaide, 1982), Lecture Notes in Math., vol. 1036, pp. 154–176.

33. , On the nonexistence of generalised Hadamard matrices, J. Statist. Plann. Inference10 (1984), no. 3, 385–396.

34. , On the nonexistence of generalised weighing matrices, Ars Combin. 17 (1984), no. A,117–132.

35. , A survey of generalised Hadamard matrices and difference matrices D(k, λ;G) withlarge k, Utilitas Math. 30 (1986), 5–29.

36. , (0, G)-designs with applications, Ph.D. thesis, University of Sydney, 1987.37. , On difference matrices, transversal designs, resolvable transversal designs and large

sets of mutually orthogonal F -squares, J. Statist. Plann. Inference 16 (1987), no. 1, 107–125.38. , GBRDs: some new constructions for difference matrices, generalised Hadamard

matrices and balanced generalised weighing matrices, Graphs Combin. 5 (1989), no. 2, 125–135.

39. , Square GBRDs over nonabelian groups, Ars Combin. 27 (1989), 40–49.40. , On the construction of n-dimensional designs from 2-dimensional designs, Aus-

tralas. J. Combin. 1 (1990), 67–81, Combinatorial Mathematics and Combinatorial Comput-ing, Vol. 1 (Brisbane, 1989).

41. , Cocyclic Hadamard matrices and relative difference sets, Ohio State Conference onGroups and Difference Sets; The Hadamard Centenary Conference, University of Wollongong,1993.

42. , On the asymptotic existence of partial complex Hadamard matrices and related com-binatorial objects, Discrete Appl. Math. 102 (2000), no. 1-2, 37–45, Coding, cryptographyand computer security (Lethbridge, AB, 1998).

43. , On a family of cocyclic Hadamard matrices, Codes and designs (Columbus, OH,2000), Ohio State Univ. Math. Res. Inst. Publ., vol. 10, de Gruyter, Berlin, 2002, pp. 187–205.

44. , On the asymptotic existence of Hadamard matrices, J. Combin. Theory Ser. A 116(2009), no. 4, 1002–1008.

45. W. de Launey and J. E. Dawson, A note on the construction of GH(4tq; EA(q)) for t = 1, 2,Australas. J. Combin. 6 (1992), 177–186.

46. , An asymptotic result on the existence of generalised Hadamard matrices, J. Combin.Theory Ser. A 65 (1994), no. 1, 158–163.

47. W. de Launey, D. L. Flannery, and K. J. Horadam, Cocyclic Hadamard matrices and dif-ference sets, Discrete Appl. Math. 102 (2000), no. 1-2, 47–61, Coding, cryptography andcomputer security (Lethbridge, AB, 1998).

48. W. de Launey and D. M. Gordon, A comment on the Hadamard conjecture, J. Combin.Theory Ser. A 95 (2001), no. 1, 180–184.

49. , A remark on Plotkin’s bound, IEEE Trans. Inf. Th. 47 (2001), no. 1.50. W. de Launey and K. J. Horadam, A weak difference set construction for higher-dimensional

designs, Des. Codes Cryptogr. 3 (1993), no. 1, 75–87.

BIBLIOGRAPHY 289

51. W. de Launey and H. Kharaghani, On the asymptotic existence of cocyclic Hadamard ma-trices, J. Combin. Theory Ser. A 116 (2009), no. 6, 1140–1153.

52. W. de Launey and D. Levin, (1,−1)-matrices with near-extremal properties, SIAM J. DiscreteMath. 23 (2009), no. 3, 1422–1440.

53. W. de Launey and J. Seberry, The strong Kronecker product, J. Combin. Theory Ser. A 66(1994), no. 2, 192–213.

54. W. de Launey and M. J. Smith, Cocyclic orthogonal designs and the asymptotic existence of

cocyclic Hadamard matrices and maximal size relative difference sets with forbidden subgroupof size 2, J. Combin. Theory Ser. A 93 (2001), no. 1, 37–92.

55. W. de Launey and R. M. Stafford, The regular subgroups of the Paley type II Hadamardmatrix, preprint.

56. , On cocyclic weighing matrices and the regular group actions of certain Paley matri-ces, Discrete Appl. Math. 102 (2000), no. 1-2, 63–101, Coding, cryptography and computersecurity (Lethbridge, AB, 1998).

57. , On the automorphisms of Paley’s type II Hadamard matrix, Discrete Math. 308(2008), no. 13, 2910–2924.

58. P. Delsarte and J.-M. Goethals, Tri-weight codes and generalized Hadamard matrices, Infor-mation and Control 15 (1969), 196–206.

59. J. F. Dillon, Variations on a scheme of McFarland for noncyclic difference sets, J. Combin.Theory Ser. A 40 (1985), no. 1, 9–21.

60. , Some REALLY beautiful Hadamard matrices, Cryptogr. Commun. 2 (2010), no. 2,271–292.

61. J. D. Dixon and B. Mortimer, Permutation groups, Graduate Texts in Mathematics, vol.163, Springer-Verlag, New York, 1996.

62. D. A. Drake, Partial λ-geometries and generalized Hadamard matrices over groups, Canad.J. Math. 31 (1979), no. 3, 617–627.

63. P. Eades, Integral quadratic forms and orthogonal designs, J. Austral. Math. Soc. Ser. A 30(1980/81), no. 3, 297–306.

64. D. L. Flannery, Calculation of cocyclic matrices, J. Pure Appl. Algebra 112 (1996), no. 2,181–190.

65. , Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. Algebra 192(1997), no. 2, 749–779.

66. D. L. Flannery and E. A. O’Brien, Computing 2-cocycles for central extensions and relativedifference sets, Comm. Algebra 28 (2000), no. 4, 1939–1955.

67. J. C. Galati, A group extensions approach to relative difference sets, J. Combin. Des. 12(2004), no. 4, 279–298.

68. H. M. Gastineau-Hills, Quasi-Clifford algebras and systems of orthogonal designs, J. Austral.Math. Soc. Ser. A 32 (1982), no. 1, 1–23.

69. A. V. Geramita and J. M. Geramita, Complex orthogonal designs, J. Combin. Theory Ser.A 25 (1978), no. 3, 211–225.

70. A. V. Geramita, J. M. Geramita, and J. S. Wallis, Orthogonal designs, Linear and MultilinearAlgebra 3 (1975/76), no. 4, 281–306.

71. A. V. Geramita and J. Seberry, Orthogonal designs, Lecture Notes in Pure and AppliedMathematics, vol. 45, Marcel Dekker Inc., New York, 1979, Quadratic forms and Hadamardmatrices.

72. P. B. Gibbons and R. Mathon, Construction methods for Bhaskar Rao and related designs,J. Austral. Math. Soc. Ser. A 42 (1987), no. 1, 5–30.

73. , Signings of group divisible designs and projective planes, Australas. J. Combin. 11(1995), 79–104.

74. P. B. Gibbons and R. A. Mathon, Group signings of symmetric balanced incomplete block de-signs, Proceedings of the Singapore conference on combinatorial mathematics and computing(Singapore, 1986), vol. 23, 1987, pp. 123–134.

75. R. E. Gilman, On the Hadamard determinant theorem and orthogonal determinants, Bull.Amer. Math. Soc. 37 (1931), 30–31.

76. J.-M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal, Canad. J. Math.19 (1967), 1001–1010.

77. D. Gorenstein, Finite groups, Chelsea Publishing Company, New York, 1980.

290 BIBLIOGRAPHY

78. S. W. Graham and I. E. Shparlinski, On RSA moduli with almost half of the bits prescribed,Discrete Appl. Math. 156 (2008), no. 16, 3150–3154.

79. The GAP group, GAP - Groups, Algorithms, and Programming, Version 4.4.9 (2006),http://www.gap-system.org.

80. J. Hadamard, Resolution d’une question relative aux determinants, Bull. des Sci. Math. 17(1893), 240–246.

81. J. Hammer and J. R. Seberry, Higher-dimensional orthogonal designs and applications, IEEE

Trans. Inform. Theory 27 (1981), no. 6, 772–779.82. D. F. Holt, The calculation of the Schur multiplier of a permutation group, Computational

group theory (Durham, 1982), Academic Press, London, 1984, pp. 307–319.83. D. F. Holt, B. Eick, and E. A. O’Brien, Handbook of computational group theory, Chapman

& Hall/CRC Press, Boca Raton, London, New York, Washington, 2005.84. W. H. Holzmann and H. Kharaghani, On the Plotkin arrays, Australas. J. Combin. 22

(2000), 287–299.85. W. H. Holzmann, H. Kharaghani, and B. Tayfeh-Rezaie, Williamson matrices up to order

59, Des. Codes Cryptogr. 46 (2008), no. 3, 343–352.86. K. J. Horadam, An introduction to cocyclic generalised Hadamard matrices, Discrete Appl.

Math. 102 (2000), no. 1-2, 115–131, Coding, cryptography and computer security (Leth-bridge, AB, 1998).

87. , Hadamard matrices and their applications, Princeton University Press, Princeton,NJ, 2007.

88. K. J. Horadam and W. de Launey, Cocyclic development of designs, J. Algebraic Combin. 2(1993), no. 3, 267–290.

89. H. Hotelling, Some improvements in weighing and other experimental techniques, Ann.Math. Statistics 15 (1944), 297–306.

90. N. Howgrave-Graham and M. Szydlo, A method to solve cyclotomic norm equations f ∗ f ,Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004,pp. 272–279.

91. B. Huppert and N. Blackburn, Finite groups. III, Grundlehren der Mathematischen Wis-senschaften, vol. 243, Springer-Verlag, Berlin, 1982.

92. Y. J. Ionin, New symmetric designs from regular Hadamard matrices, Electron. J. Combin.5 (1998), Research Paper 1, 8 pp. (electronic).

93. , A technique for constructing symmetric designs, Des. Codes Cryptogr. 14 (1998),no. 2, 147–158.

94. , Building symmetric designs with building sets, Des. Codes Cryptogr. 17 (1999),no. 1-3, 159–175.

95. , Symmetric subdesigns of symmetric designs, J. Combin. Math. Combin. Comput.29 (1999), 65–78.

96. , Applying balanced generalized weighing matrices to construct block designs, Elec-tron. J. Combin. 8 (2001), no. 1, Research Paper 12, 15 pp. (electronic).

97. I. M. Isaacs, Algebra: a graduate course, Brooks/Cole, Pacific Grove, 1994.98. N. Ito, Note on Hadamard matrices of type Q, Studia Sci. Math. Hungar. 16 (1981), no. 3-4,

389–393.99. , Note on Hadamard groups of quadratic residue type, Hokkaido Math. J. 22 (1993),

no. 3, 373–378.100. , On Hadamard groups, J. Algebra 168 (1994), no. 3, 981–987.101. , On Hadamard groups, II, J. Algebra 169 (1994), no. 3, 936–942.

102. , On Hadamard groups III, Kyushu J. Math. 51 (1997), no. 3, 369–379.103. Z. Janko, H. Kharaghani, and V. D. Tonchev, Bush-type Hadamard matrices and symmetric

designs, J. Combin. Des. 9 (2001), no. 1, 72–78.104. , The existence of a Bush-type Hadamard matrix of order 324 and two new infinite

classes of symmetric designs, Des. Codes Cryptogr. 24 (2001), no. 2, 225–232.105. D. Jungnickel, On difference matrices, resolvable transversal designs and generalized

Hadamard matrices, Math. Z. 167 (1979), no. 1, 49–60.106. W. M. Kantor, Automorphism groups of Hadamard matrices, J. Combinatorial Theory 6

(1969), 279–281.107. , Symplectic groups, symmetric designs, and line ovals, J. Algebra 33 (1975), 43–58.

BIBLIOGRAPHY 291

108. G. Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Se-ries, vol. 2, The Clarendon Press Oxford University Press, New York, 1987.

109. H. Kharaghani, An asymptotic existence result for orthogonal designs, Combinatorics ad-vances (Tehran, 1994), Math. Appl., vol. 329, Kluwer Acad. Publ., Dordrecht, 1995, pp. 225–233.

110. , On the twin designs with the Ionin-type parameters, Electron. J. Combin. 7 (2000),Research Paper 1, 11 pp. (electronic).

111. H. Kharaghani and J. Seberry, Regular complex Hadamard matrices, Proceedings of theNineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg,MB, 1989), vol. 75, 1990, pp. 187–201.

112. H. Koch, Number theory. Algebraic numbers and functions, Graduate Studies in Mathemat-ics, vol. 24, American Mathematical Society, Providence, RI, 2000.

113. T. Y. Lam and K. H. Leung, On vanishing sums of roots of unity, J. Algebra 224 (2000),no. 1, 91–109.

114. J. S. Leon, An algorithm for computing the automorphism group of a Hadamard matrix, J.Comb. Theory, Ser. A 27 (1979), no. 3, 289–306.

115. C. Mackenzie and J. Seberry, Maximal ternary codes and Plotkin’s bound, Ars Combin. 17(1984), no. A, 251–270.

116. V. C. Mavron, T. P. McDonough, and C. A. Pallikaros, A difference matrix constructionand a class of balanced generalized weighing matrices, Arch. Math. (Basel) 76 (2001), no. 4,259–264.

117. V. C. Mavron and V. D. Tonchev, On symmetric nets and generalized Hadamard matricesfrom affine designs, J. Geom. 67 (2000), no. 1-2, 180–187, Second Pythagorean Conference(Pythagoreion, 1999).

118. R. L. McFarland, Hadamard difference sets in abelian groups of order 4p2, Mitt. Math. Sem.Giessen. 192 (1989), 1–70.

119. , Sub-difference sets of Hadamard difference sets, J. Combin. Theory Ser. A 54 (1990),no. 1, 112–122.

120. B. McKay, Practical graph isomorphism, Congressus Numerantium 30 (1981), 45–87.121. A. C. Mukhopadhyay, Generalized weighing matrices, SGDDs possessing dual property and

related configurations, Sankhya Ser. A 54 (1992), no. Special Issue, 291–298, Combinatorialmathematics and applications (Calcutta, 1988).

122. R. C. Mullin, A note on balanced weighing matrices, Combinatorial mathematics, III (Proc.Third Australian Conf., Univ. Queensland, St. Lucia, 1974), Springer, Berlin, 1975, pp. 28–41. Lecture Notes in Math., Vol. 452.

123. R. C. Mullin and R. G. Stanton, Balanced weighing matrices and group divisible designs,Utilitas Math. 8 (1975), 303–310.

124. , Group matrices and balanced weighing designs, Utilitas Math. 8 (1975), 277–301.125. W. Nickel, Central extensions of polycyclic groups, Ph.D. thesis, Australian National Uni-

versity, 1993.

126. P. O Cathain and M. Roder, The cocyclic Hadamard matrices of order less than 40, Des.Codes Cryptogr. 58 (2011), no. 1, 73–88.

127. D. Z. Dokovic, Periodic complementary sets of binary sequences, Int. Math. Forum 4 (2009),no. 13-16, 717–725.

128. R. E. A. C. Paley, On orthogonal matrices, J. Math. Phys. 12 (1933), 311–320.129. A. A. I. Perera and K. J. Horadam, Cocyclic generalised Hadamard matrices and central

relative difference sets, Des. Codes Cryptogr. 15 (1998), no. 2, 187–200.130. A. Pott, Finite geometry and character theory, Lecture Notes in Math., vol. 1601, Springer-

Verlag, Berlin, 1995.131. D. P. Rajkundlia, Some techniques for constructing new infinite families of incomplete block

designs, Ph.D. thesis, Queens University, Kingston, Canada, 1978.

132. , Some techniques for constructing infinite families of BIBDs, Discrete Math. 44(1983), no. 1, 61–96.

133. D. K. Ray-Chaudhuri and Q. Xiang, New necessary conditions for abelian Hadamard differ-ence sets, J. Statist. Plann. Inference 62 (1997), 69–79.

134. D. J. S. Robinson, A course in the theory of groups, second ed., Graduate Texts in Mathe-matics, vol. 80, Springer-Verlag, New York, 1996.

292 BIBLIOGRAPHY

135. D. G. Sarvate and J. Seberry, Group divisible designs, GBRSDS and generalized weighingmatrices, Util. Math. 54 (1998), 157–174.

136. P. J. Schellenberg, A computer construction for balanced orthogonal matrices, Proceedingsof the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing(Florida Atlantic Univ., Boca Raton, Fla., 1975) (Winnipeg), Utilitas Math., 1975, pp. 513–522. Congressus Numerantium, No. XIV.

137. S. T. Schibell and R. M. Stafford, private communications, 1993, 2011.

138. B. Schmidt, Cyclotomic integers and finite geometry, J. Amer. Math. Soc. 12 (1999), no. 4,920–952.

139. , Williamson matrices and a conjecture of Ito’s, Des. Codes Cryptogr. 17 (1999),no. 1-3, 61–68.

140. , Towards Ryser’s conjecture, European Congress of Mathematics, Vol. I (Barcelona,2000), Progr. Math., vol. 201, Birkhauser, Basel, 2001, pp. 533–541.

141. , Characters and cyclotomic fields in finite geometry, Lecture Notes in Math., vol.1797, Springer-Verlag, Berlin, 2002.

142. J. Seberry, Some remarks on generalised Hadamard matrices and theorems of Rajkundlia onSBIBDs, Combinatorial mathematics, VI (Proc. Sixth Austral. Conf., Univ. New England,Armidale, 1978), Lecture Notes in Math., vol. 748, Springer, Berlin, 1979, pp. 154–164.

143. , A construction for generalized Hadamard matrices, J. Statist. Plann. Inference 4(1980), no. 4, 365–368.

144. J. Shawe-Taylor, Coverings of complete bipartite graphs and associated structures, DiscreteMath. 134 (1994), no. 1-3, 151–160, Algebraic and topological methods in graph theory(Lake Bled, 1991).

145. P. J. Shlichta, Higher dimensional Hadamard matrices, IEEE Trans. Inform. Theory 25(1979), no. 5, 566–572.

146. S. S. Shrikhande, Generalized Hadamard matrices and orthogonal arrays of strength two,Canad. J. Math. 16 (1964), 736–740.

147. C. C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics andits Applications, vol. 48, Cambridge University Press, Cambridge, 1994.

148. D. J. Street, Generalized Hadamard matrices, orthogonal arrays and F -squares, Ars Combin.

8 (1979), 131–141.149. J. J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign successions, and

tesselated pavements in two or more colours, with applications to Newton’s rule, ornamentaltile-work, and the theory of numbers, Phil. Mag. 34 (1867), 461–475.

150. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from orthogonaldesigns, IEEE Trans. Inform. Theory 45 (1999), no. 5, 1456–1467.

151. R. J. Turyn, Sequences with small correlation, Error Correcting Codes (Proc. Sympos. Math.Res. Center, Madison, Wis., 1968), John Wiley, New York, 1968, pp. 195–228.

152. , Complex Hadamard matrices, Combinatorial Structures and their Applications(Proc. Calgary Internat. Conf., Calgary, Alta., 1969), Gordon and Breach, New York, 1970,pp. 435–437.

153. , On C-matrices of arbitrary powers, Canad. J. Math. 23 (1971), 531–535.154. , An infinite class of Williamson matrices, J. Combinatorial Theory Ser. A 12 (1972),

319–321.155. , A special class of Williamson matrices and difference sets, J. Combin. Theory Ser.

A 36 (1984), no. 1, 111–115.156. Jennifer Seberry Wallis, On the existence of Hadamard matrices, J. Combinatorial Theory

Ser. A 21 (1976), no. 2, 188–195.157. J. Williamson, Hadamard’s determinant theorem and the sum of four squares, Duke Math.

J. 11 (1944), 65–81.158. R. M. Wilson and Q. Xiang, Constructions of Hadamard difference sets, J. Combin. Theory

Ser. A 77 (1997), no. 1, 148–160.

159. W. Wolfe, Rational quadratic forms and orthogonal designs, Number theory and algebra,Academic Press, New York, 1977, pp. 339–348.

160. , Limits on pairwise amicable orthogonal designs, Canad. J. Math. 33 (1981), no. 5,1043–1054.

161. M. Y. Xia, Some infinite classes of special Williamson matrices and difference sets, J.Combin. Theory Ser. A 61 (1992), no. 2, 230–242.

BIBLIOGRAPHY 293

162. Q. Xiang and Y. Q. Chen, On Xia’s construction of Hadamard difference sets, Finite Fieldsand Their Applications 2 (1996), 87–95.

163. M. Yamada, Hadamard matrices of generalized quaternion type, Discrete Math. 87 (1991),no. 2, 187–196.

164. K. Yamamoto, On a generalized Williamson equation, Finite and infinite sets, Vol. I, II(Eger, 1981), Colloq. Math. Soc. Janos Bolyai, vol. 37, North-Holland, Amsterdam, 1984,pp. 839–850.

165. Y. X. Yang, X. X. Niu, and C. Q. Xu, Theory and applications of higher-dimensionalHadamard matrices, 2nd ed., CRC Press, Boca Raton, 2010.

Index

A ∧B, 41

A⊗B, 41

A(−1), 127

Bn(G,C), 242

CG(H), 28

E(f), 145

G′, 31G(i), 31

H �G, 36

Hn(G,C), 242

H2(G), 245

Iv, 7

Jv, 7

K � L, L �K, 32

K �C L, K � L, 32

K · L, 32K �C L, K � L, 32

NG(H), 28

PA, 54

P f(x,c)

, 157

R-, C-distinct, 103

Sx, Tx, 45

X∗, 7, 65Z(G), 28

Zn(G,C), 242

[A,B], 31

[a, b], 31

ΓL(V ), 43

ΓL(n,F), 43Λ, 7

Λ-cocycle, 163

pure, 163

Λ-indexing group, 163

Λ-orthogonal, 7

Λ1 � Λ2, 74

Λ1 �Z Λ2, 191

Λ1 ⊗ Λ2, 73

ΦΛ, 56

ΠΛ, 55

ΠcolΛ , 54

ΠrowΛ , 54

ΨΛ, 80

ΘR,C , 110

α(∗), 128≈, 60≈Λ, 55δ(Λ), 51λ(Δ), 51C, 7F×, 39N, 18Q, 11Z, 19Zd, 30C(D), 164C(Λ), 164CE,C(ι, π, τ) etc., 147E(D), 108EK,L(D), 108AGL(k,F), 101Alt(n), 36Aut(D), 99AD, 114BGW(v, k, λ;G), 22BW(v, k, λ), 13CGH(n; k), 18CGW(v, k;m), 19Cn, 34DT(G), 242DTfree(G), DTtor(G), 242Diag(n,C), 42D2n, 34

E(D), 168En, 34Frat(G), 32GDD(vm, k, λ;m), 116GF(pr), 40GH(n;G), 20GL(V ), 43GL(n,R), 42GL(n, q), 43GW(v, k;G), 23GramR(Λ), 71Hol(G), 33Mat(n,R), 41Mon(n,C), 42OD(n; a1, . . . , ar), 14

295

296 INDEX

PCD(Λ), 8PCD(v,Λ), 8PCDn(Λ), 135PGL(n,F), 43PSL(n, q), 43PermAut(D), 105Perm(n), 41

Q4t, 34SBIBD(v, k, λ), 9SL(n,F), 43Sym(Ω), 35Sym(n), 36W(n, k), 12wt(X), df(X), 72wt(Λ), df(Λ), 73∂ρ, 147�nΛ, 90f1 × f2, 150fι,τ , 146x y, 210

action, 35k-transitive, 38faithful, 36induced, 38

normal, 197regular, 37semiregular, 37transitive, 37

algebra, 39algebraic design theory, 1alphabet, 7ambient ring, 63

with central group, 162with row group and column group, 63

amicable, 16anti-, 16

anisotropic vector, 260array

Λ-equivalent, 54f -developed, 140n-dimensional, 135

section, 135cocyclic, 161cocyclic form, 161Goethals-Seidel, 89group-developed, 120non-degenerate, 72

permutation equivalent, 61Williamson, 86

associateG-associate, 127f -associate, 177cocyclic, 177

auxiliary matrices, 234

center, 28central isomorphism, 150centralizer, 28

centrally regularembedding, 172subgroup, 171

character, 186principal, 186quadratic, 205

coboundary, 147

n-, 242cocycle, 139

n-, 242almost symmetric, 252binary, 201collection, 154composition, 190fully free, 246fully torsion, 246normalized, 145product, 150symmetric, 155trivial, 145

cocycle identity, 135cohomologous, 147cohomology

class, 147group, 242

collapsible function, 136combinatorial design theory, 1commutator, 31complement, 32

normal p-, 182

complementary pairreal, complex, 223

concordant system of orthogonal designs,199

coset, 28, 40

designassociated, 114expanded, 108group divisible, 116lesser expanded, 108orthogonal, 14symmetric balanced incomplete block, 9

design set, 51full, 52

development function, 135Λ-row-invariant, 140Λ-suitable, 141

abelian, 141normalized, 140

development table, 239free, 242torsion, 242

difference set, 124

equivalence operationconjugate, 80global, 56local, 54

INDEX 297

expanded design

block matrix form, 108plug-in form, 109

abelian, 109extension, 145

central, 145

canonical, 145split, 32

extension function, 137

field, 39Galois, 40

of fractions, 39primitive element, 40

Fourier inversion formula, 187Frobenius map, 41

Golay complementary sequence, 280Gram Property, 71

Grammian, 42group, 27

p-, 27abelian, 27

affine, 101alternating, 36automorphism, 29

cyclic, 27dicyclic, 34

dihedral, 34elementary abelian, 31

exponent, 27extraspecial, 34

finitely generated, 29free, 29

rank, 29

free abelian, 29rank, 29

generalized quaternion, 34generating set, 29

involution, 27metacyclic, 33

nilpotent, 31order, 27permutation, 36

degree, 36polycyclic, 152

product, 32quotient, 29

simple, 28solvable, 31

symmetric, 35group ring, 39group ring equation, 127, 178

Hadamard cocycle, 195

pure, 195Hadamard group, 195

holomorph, 33homomorphism

of groups, 29of monoids, 35

of rings, 40Hopf’s formula, 249

ideal, 40

maximal, 40prime, 40

inflation, 243integral domain, 39isomorphism, 29

permutation, 36

kernel, 29, 36, 40

linear group, 43degree, 43irreducible, 43

lines, 204half-, 204

matrixω-cyclic, 228cofactor, 41

conference, 13Gram, 42Hadamard, 11

complex, 16complex generalized, 18

generalized, 20incidence, 9monomial, 42negacyclic, 228permutation, 41

regular, 17, 226weighing, 12

balanced, 13complex generalized, 19

generalized, 23Williamson-like, 200

monoid, 34monoid ring, 39

near field, 209norm equation, 189normal closure, 30normalizer, 28

orbit, 36orthogonality set, 7

α-transposable, 78conjugate transposable, 21, 78

Gram complete, 73Gram completion, 73irredundant, 52Kronecker product, 73non-degenerate, 72

self-dual, 78transposable, 77trivial, 50

298 INDEX

pairwise combinatorial design, 8automorphism group, 99cocyclic, 161

central short exact sequence of, 164cocycle of, 163cocyclic form of, 167extension group of, 165

permutation automorphism group, 105proper n-dimensional, 135

Paleyconference matrix, 205type I Hadamard matrix, 206type II Hadamard matrix, 207

polynomial ring, 40presentation

finite, 30polycyclic, 152

consistent, 152primary-invariant form, 30product

central, 32direct, 32Hadamard, 41Kronecker, 41semidirect, 32wreath, 36

regular embedding, 122relative difference set, 125

central, 181forbidden subgroup, 125normal, 126

representation, 44faithful, 44monomial, 157permutation, 35

similarity, 38ring, 39

characteristic, 39involution, 42involutory, 42quotient, 40unit, 39

row and column operationselementary, 54

Schurcomplement, 248cover, 249multiplier, 245

Schur-Zassenhaus theorem, 33self-conjugate

ideal, 187integer, 187

semilinear group, 43semilinear transformation, 43short exact sequence, 146

central, 146canonical, 146

Singer cycle, 43stabilizer, 36subgroup, 27

characteristic, 29conjugate, 28derived, 31Frattini, 32

index, 28normal, 28Sylow, 28

subring, 39substitution scheme, 87Sylvester Hadamard matrix, 257system of imprimitivity, 39

torsion element, 27torsion subgroup, 30torsion-invariant form, 30transference, 93transgression, 247translation, 122, 171transversal, 28

map, 146

zero aperiodic autocorrelation, 279zero divisor, 39

Titles in This Series

175 Warwick de Launey and Dane Flannery, Algebraic design theory, 2011

174 Lawrence S. Levy and J. Chris Robson, Hereditary Noetherian prime rings andidealizers, 2011

173 Sariel Har-Peled, Geometric approximation algorithms, 2011

172 Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon,The classification of finite simple groups: Groups of characteristic 2 type, 2011

171 Leonid Pastur and Mariya Shcherbina, Eigenvalue distribution of large randommatrices, 2011

170 Kevin Costello, Renormalization and effective field theory, 2011

169 Robert R. Bruner and J. P. C. Greenlees, Connective real K-theory of finite groups,2010

168 Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko, Algebras, rings andmodules: Lie algebras and Hopf algebras, 2010

167 Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebra andPoisson geometry, 2010

166 Kyung Bai Lee and Frank Raymond, Seifert fiberings, 2010

165 Fuensanta Andreu-Vaillo, Jose M. Mazon, Julio D. Rossi, and J. JulianToledo-Melero, Nonlocal diffusion problems, 2010

164 Vladimir I. Bogachev, Differentiable measures and the Malliavin calculus, 2010

163 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, JamesIsenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow:Techniques and applications, Part III: Geometric-analytic aspects, 2010

162 Vladimir Maz′ya and Jurgen Rossmann, Elliptic equations in polyhedral domains,2010

161 Kanishka Perera, Ravi P. Agarwal, and Donal O’Regan, Morse theoretic aspectsof p-Laplacian type operators, 2010

160 Alexander S. Kechris, Global aspects of ergodic group actions, 2010

159 Matthew Baker and Robert Rumely, Potential theory and dynamics on theBerkovich projective line, 2010

158 D. R. Yafaev, Mathematical scattering theory: Analytic theory, 2010

157 Xia Chen, Random walk intersections: Large deviations and related topics, 2010

156 Jaime Angulo Pava, Nonlinear dispersive equations: Existence and stability of solitaryand periodic travelling wave solutions, 2009

155 Yiannis N. Moschovakis, Descriptive set theory, 2009

154 Andreas Cap and Jan Slovak, Parabolic geometries I: Background and general theory,2009

153 Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques inspectral analysis, 2009

152 Janos Pach and Micha Sharir, Combinatorial geometry and its algorithmicapplications: The Alcala lectures, 2009

151 Ernst Binz and Sonja Pods, The geometry of Heisenberg groups: With applications insignal theory, optics, quantization, and field quantization, 2008

150 Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang, Finite dimensionalalgebras and quantum groups, 2008

149 Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, 2008

148 Patrick Dehornoy with Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Orderingbraids, 2008

147 David J. Benson and Stephen D. Smith, Classifying spaces of sporadic groups, 2008

146 Murray Marshall, Positive polynomials and sums of squares, 2008

145 Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin, Simple groups of finiteMorley rank, 2008

TITLES IN THIS SERIES

144 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, JamesIsenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow:Techniques and applications, Part II: Analytic aspects, 2008

143 Alexander Molev, Yangians and classical Lie algebras, 2007

142 Joseph A. Wolf, Harmonic analysis on commutative spaces, 2007

141 Vladimir Maz′ya and Gunther Schmidt, Approximate approximations, 2007

140 Elisabetta Barletta, Sorin Dragomir, and Krishan L. Duggal, Foliations inCauchy-Riemann geometry, 2007

139 Michael Tsfasman, Serge Vladut, and Dmitry Nogin, Algebraic geometric codes:Basic notions, 2007

138 Kehe Zhu, Operator theory in function spaces, 2007

137 Mikhail G. Katz, Systolic geometry and topology, 2007

136 Jean-Michel Coron, Control and nonlinearity, 2007

135 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, JamesIsenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow:Techniques and applications, Part I: Geometric aspects, 2007

134 Dana P. Williams, Crossed products of C∗-algebras, 2007

133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006

132 J. P. May and J. Sigurdsson, Parametrized homotopy theory, 2006

131 Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, 2006

130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds inEuclidean spaces, 2006

129 William M. Singer, Steenrod squares in spectral sequences, 2006

128 Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu.Novokshenov, Painleve transcendents, 2006

127 Nikolai Chernov and Roberto Markarian, Chaotic billiards, 2006

126 Sen-Zhong Huang, Gradient inequalities, 2006

125 Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy Transform,2006

124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006

123 Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, NitinNitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck’s FGAexplained, 2005

122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities andasymptotic methods, 2005

121 Anton Zettl, Sturm-Liouville theory, 2005

120 Barry Simon, Trace ideals and their applications, 2005

119 Tian Ma and Shouhong Wang, Geometric theory of incompressible flows with

applications to fluid dynamics, 2005

118 Alexandru Buium, Arithmetic differential equations, 2005

117 Volodymyr Nekrashevych, Self-similar groups, 2005

116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005

115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005

114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005

113 William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith,Homotopy limit functors on model categories and homotopical categories, 2004

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/.

SURV/175

www.ams.orgAMS on the Web

For additional informationand updates on this book, visit

www.ams.org/bookpages/surv-175

Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs—new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.

Phot

ogra

ph c

ourt

esy

of I

one

Rum

mer

y