Table Of ContentJennifer Seberry
Orthogonal
Designs
Hadamard Matrices, Quadratic Forms
and Algebras
Orthogonal Designs
Jennifer Seberry
Orthogonal Designs
Hadamard Matrices, Quadratic Forms
and Algebras
123
Jennifer Seberry
Schoolof Computing andInformation
Technology
University of Wollongong
Wollongong, NSW
Australia
ISBN978-3-319-59031-8 ISBN978-3-319-59032-5 (eBook)
DOI 10.1007/978-3-319-59032-5
LibraryofCongressControlNumber:2017945720
©SpringerInternationalPublishingAG2017
Originalpublication:“OrthogonalDesigns:Hadamardmatricesandquadraticforms”-A.V.Geramita,
J.Seberry.1979.MarcelDeckkerPress-Taylor&Francis,US.
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This book is dedicated, with the greatest
respect, to Anthony V Geramita who has
now gone to the Erdos’ great SF. I thank
Joan Geramita, his friend, colleague and
heir, for permission to use Tony’s work in
this updated volume.
Preface
Problems concerned with the structure and existence of various kinds of
matriceswithelementsfrom0,1,-1,forexample,Hadamardmatricesandtheir
generalization to weighing matrices, have long been of interest to workers in
combinatoricsandalsoappliedstatisticians,codingtheorists,signalprocessors
and other applied mathematicians. A first volume, “Orthogonal Designs:
quadratic forms and Hadamard matrices” (Ed 1), was written jointly by
Anthony V. Geramita and Jennifer Seberry and published by Marcel Dekker
in 1979, but never reprinted. This 1979 volume, devoted to a ground-breaking
approach,illuminatedtheconnectionsbetweenthesevariouskindsofmatrices
and exposed new connections with several other areas of mathematics. The
current volume, “Orthogonal Designs: Hadamard matrices, quadratic forms
and algebras”, is the revision and update of the initial volume created using
research theses and papers written in the intervening years. This more recent
researchhasledtonewideasformanyareasofmathematics,signalprocessing
and non-deterministic computer programming in computational mathematics.
These approaches are through the investigation of orthogonal designs: roughly
speaking, special matrices with indeterminate entries.
Originally this subject had been discussed in our research papers and those
of our colleagues and students. The discovery of the intimate relationship
between orthogonal designs and rational quadratic forms had not appeared
in print before 1973. The finding of numerous constructions and interesting
objects that appeared fundamental to the study of Hadamard matrices (and
their generalizations) finally prompted Geramita and Seberry to look afresh
at the work that had already appeared. They recast their work and their
collaborators and students in the light of their new discoveries. This present
updated and new work continues the previous book and introduces more
recent material by collaborators, colleagues and students. It leads to new
algebras, techniques and existence results.
As will be clear in the text, orthogonal designs is a heavy “borrower” of
mathematics.Thereaderwillfindususingresultsfrom,forexample,algebraic
number theory, quadratic forms, difference sets, representation theory, coding
vii
viii Preface
theory, finite geometry, elementary number theory, cyclotomy, the theory of
computationandsignalprocessing.Thereaderisnotexpectedtobeconversant
with all these areas; the material is presented in such a way that even the
novice to these areas will understand why and how we intend to use the
results stated, even if the proofs in some cases remain a mystery. In those
cases where detailed explanation would take too long, references are given so
the interested reader can fill out their background. Thus the original volume
and this volume can be profitably read both by experts and by people new to
this area of discrete mathematics and combinatorics.
To dispel any notion that this book closes the area for further research,
many problems are highlighted, all unsolved, and directions in which further
research is possible are suggested. These problems vary in depth: some are
seemingly very simple, others are major.
Some comments on how this volume is organized: the organization is,
in part, directed by the Janus-like features of the study. In the first three
chapters, which largely remain untouched and are heavily underpinned by the
farsightedworkofAnthonyV.Geramita,thenatureoftheproblemathandis
described,andsomeremarksmadeontheingredientsofasolution.Aftersome
preliminaries,aratherdeepforayismadeintothealgebrasideofthequestion.
In broad terms, the algebra there described allows us to identify the first set
of non-trivial necessary conditions on the problem of existence of orthogonal
designs. Chapter 4 concentrates, and with the necessary conditions as a guide,
onattemptstosatisfytheseconditions.Manydifferentmethodsofconstruction
are described and analysed. Chapter 5 focusses on one of these construction
methods and analyses it in detail, both algebraically and combinatorially.
Hereagain,theinterplaybetweenclassicalalgebraandcombinatoricsisshown
to have striking consequences. Chapter 6 deeply studies two construction
methods introduced but not analysed in the original book. The result is new
algebraswhichhavebeendevelopedtoencompassthesecombinatorialconcepts.
Chapter7deviatestogivesomeofthetheoryandexistenceresultsforareasof
number theory and discrete applied mathematics which have proved, over the
pastfortytofiftyyears,tohavebeensomewhatforgottenbythosenotstudying
orthogonal designs. In Chapter 8 a very strong non-existence theorem for
orthogonaldesignsisproved.The“AsymptoticHadamardExistenceTheorem”
and related wonderful asymptotic consequences and questions, which are
central to Chapter 9, are due to a number of authors. Chapter 10 reminds
us that we have not finished with number theoretic consequences and other
combinatorial features of orthogonal designs by commencing the study of
non-real fields. Finally, in the Appendices, we tabulate numerous calculations
we have made in specific orders.
Preface ix
Acknowledgements
No book this size could be the child of one or two parents. Many people are
owedadebtofdeepgratitude.FortheoriginalbookDanShapiro(OhioState)
helped immensely with his work on similarities. The 1970’s students Peter
Robinson(ANU,Canberra),PeterEades(ANU,Canberra)andWarrenWolfe
(Queens’,Kingston)hadmajorinput;the1980’sstudentsofSeberry,Deborah
J. Street (Sydney), Warwick de Launey (Sydney) and Humphrey Gastineau-
Hills (Sydney), and the 2000’s students Chung Le Tran (Wollongong) and
YingZhao(Wollongong)havecontributedconspicuouslytothevolume.Many,
many colleagues such as Marilena Mitrouli (Athens), Christos Koukouvinos
(NTUA, Athens) and his students Stelios Georgiou and Stella Stylianou,
Hadi Kharghani (Lethbridge), Wolf Holzmann (Lethbridge), Rob Craigen
(Winnipeg), Ilias Kotsireas (WLU), Dragomir Ðoković (Waterloo), Sarah
Spence Adams (Olin, Boston), Behruz Tayfeh-Rezaie(IPM, Tehran) and
Ebrahim Ghaderpour (York, Toronto) have helped shape and correct our
work. I acknowledge that at times I might have written utter rubbish without
their knowledge and thoughts. I leave writing this book knowing it is an
unfinished work, knowing that I have included many errors large and small.
I acknowledge the generous proofing help for the original volume by Joan
Geramita and Chuck Weibel. I acknowledge the LaTeXing proofing, and
exceptional research assistance given me over the past ten years by Max
Norden. I came to today with the help of the University of Wollongong
Library and the provision of an office to me as Emeritus Professor.
From the original preface “Finally, a word of our experience writing this
book. We approached the problems herein from quite different backgrounds.
It became evident that Geramita was more interested in the connections with
algebra, while Seberry was more interested in actually making the objects
that could be made. The inevitable tensions that arise from these differing
viewpoints (will), we hope, make this book more interesting to read.”
Wollongong, Australia Jennifer Seberry
November 2016
Contents
1 Orthogonal Designs....................................... 1
1.1 Hurwitz-Radon families ................................. 2
2 Non-existence Results .................................... 7
2.1 Weighing Matrices...................................... 7
2.2 Odd Order ............................................ 8
2.3 Algebraic Problem...................................... 13
2.4 Orthogonal Designs’ Algebraic Problem ................... 13
2.5 Geramita-Verner Theorem Consequences .................. 15
3 Algebraic Theory of Orthogonal Designs.................. 19
3.1 Generalities on Quadratic and Bilinear Forms .............. 19
3.2 The Matrix Formulation................................. 21
3.3 Mapping Between Bilinear Spaces ........................ 22
3.4 New Spaces From Old................................... 23
3.5 Bilinear Spaces Classification Theorems ................... 24
3.6 Classification of Quadratic Forms Over Q.................. 25
3.7 The Similarities of a Bilinear Space ....................... 30
3.8 Linear Subspaces of Sim(V) ............................. 31
3.9 Relations Between Rational Families in the Same Order ..... 36
3.10 Clifford Algebras ....................................... 37
3.11 Similarity Representations ............................... 38
3.12 Some Facts About Positive Definite Forms Over Q.......... 40
3.13 Reduction to Powers of 2 ................................ 43
3.14 Orders 4 and 8......................................... 46
3.14.1 Order 4 ......................................... 46
3.14.2 Order 8 ......................................... 48
3.15 Order 16 .............................................. 51
3.15.1 Case 1: 9-member rational families. ................. 53
3.15.2 Case 2: 7-member rational families. ................. 53
3.15.3 Case 3: 8-member rational families. ................. 54
xi
xii Contents
3.16 Order 32 .............................................. 56
3.17 Solution of the Algebraic Problem ........................ 57
3.18 Combining Algebra with Combinatorics ................... 59
3.18.1 Alert ........................................... 61
4 Orthogonal Designs Constructed via Plug-in Matrices .... 63
4.1 Introduction ........................................... 63
4.2 Some Orthogonal Designs Exist .......................... 63
4.3 Some Basic Matrix Results .............................. 68
4.3.1 Supplementary Difference Sets, their Incidence
Matrices and their Uses as Suitable Matrices......... 74
4.4 Existence of Weighing Matrices .......................... 76
4.5 Constructions for W(h,h) and W(h,h−1)................. 82
4.6 Using Circulants–Goethals-Seidel Array and Kharaghani Array 89
4.7 Constraints on construction using circulant matrices ........ 95
4.8 Eades’ Technique for Constructing Orthogonal Designs...... 96
4.9 Some Arrays for Eight Circulants......................... 107
4.10 Amicable Sets and Kharaghani Arrays .................... 110
4.11 Construction using 8 Disjoint Matrices .................... 111
4.11.1 Hadamard Matrices............................... 115
4.12 Baumert-Hall Arrays.................................... 117
4.13 Plotkin Arrays ......................................... 124
4.13.1 Kharaghani’s Plotkin arrays ....................... 126
4.14 More Specific Constructions using Circulant Matrices ....... 126
4.15 Generalized Goethals-Seidel Arrays ....................... 129
4.15.1 Some Infinite Families of Orthogonal Designs ........ 133
4.15.2 Limitations...................................... 134
4.16 Balanced Weighing Matrices ............................. 134
4.16.1 Necessary Conditions for the Existence of Balanced
Weighing Matrices................................ 135
4.16.2 Construction Method for Balanced Weighing Designs . 136
4.16.3 Regular Balanced Weighing Matrices ............... 139
4.16.4 Application of the Frobenius Group Determinant
Theorem to Balanced Weighing Matrices ............ 141
4.16.5 Balanced Weighing Matrices with v≤25 ............ 143
4.16.6 No Circulant Balanced Weighing Matrices
BW(v,v−1) Based on (v,v−1,v−2) Configurations . 144
4.17 Negacyclic Matrices..................................... 148
4.17.1 Constructions.................................... 152
4.17.2 Applications ..................................... 153
4.17.3 Combinatorial Applications........................ 154
