Designs and their Codes

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Overview

Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications, together with a fairly thorough general introduction to both design theory and coding theory, developing the relationship between the two areas. The first half of the book contains general background material in design theory, including symmetric designs and designs from affine and projective geometries, and in coding theory, including coverage of most of the important classes of linear codes. In particular the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems. The book is aimed at mathematicians working in either coding theory or combinatorics - or related areas of algebra. The book is, however, also designed to be used by non-specialists and can be used by those graduate students or computer scientists who may be working in these areas.
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Editorial Reviews

From the Publisher
"...a valuable resource for researchers in either finite geometries or coding theory as well as for algebraists who want to learn about this lively, growing area." Vera Pless, Mathematical Reviews

"...the relationship between the two subjects is very much a two-way channel, and the book is a mine of useful information from whichever direction one approaches it....a useful compilation of material which, together with the extensive bibliography, will prove useful to anyone whose research impinges on these topics." N.L. Biggs

"...speaks to the tremendous influence the plane of order ten has subsequently had on the analysis and classification of designs in a much broader context than projective planes...a welcome addition to a very exciting and relatively new application of an established discipline to combinatorics...a truly fascinating and useful book. It belongs on the shelves of all those who wish to be current on the state of design theory and who are seeking interesting problems in the field to pursue." M.A. Wertheimer, Bulletin of the American Mathematical Society

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Product Details

  • ISBN-13: 9780521413619
  • Publisher: Cambridge University Press
  • Publication date: 1/28/2004
  • Series: Cambridge Tracts in Mathematics Series , #103
  • Pages: 364
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.94 (d)

Table of Contents

Preface
1 Designs
1.1 Introduction 1
1.2 Basic definitions 1
1.3 Related structures 13
1.4 Ranks of incidence matrices 17
1.5 Arcs and ovals 19
1.6 Block's theorem 20
2 Codes
2.1 Introduction 25
2.2 Linear codes 31
2.3 Parity-check matrices 35
2.4 Codes from designs 41
2.5 Hamming codes 55
2.6 Cyclic codes 60
2.7 Transforms 66
2.8 Cyclic codes through roots 71
2.9 MDS codes 74
2.10 Quadratic-residue codes 76
2.11 Weight enumerators 81
3 The geometry of vector spaces
3.1 Introduction 89
3.2 Projective geometry 90
3.3 Affine geometry 96
3.4 Designs from geometries 99
3.5 The cross ratio 101
3.6 Desargues' theorem 103
3.7 Quadrics and hermitian varieties 104
4 Symmetric Designs
4.1 Introduction 117
4.2 The parameters 118
4.3 Automorphisms 122
4.4 Difference sets 123
4.5 Group algebras 129
4.6 The codes and multiplier theorem 132
5 The standard geometric codes
5.1 Introduction 139
5.2 The Reed-Muller codes 140
5.3 Geometries and Reed-Muller codes 145
5.4 Generalized Reed-Muller codes 152
5.5 Dimensions and minimum weights 160
5.6 The geometric codes 167
5.7 The subfield subcodes 177
5.8 Summation formulas for the p-rank 193
6 Codes from planes
6.1 Introduction 199
6.2 Projective to affine and back 201
6.3 The codes 203
6.4 The minimum-weight vectors 209
6.5 Central collineations 214
6.6 Other geometric codewords 218
6.7 Hermitian unitals 225
6.8 Translation planes 228
6.9 Tame planes and a rigidity theorem 235
6.10 Derivations 238
6.11 Ovals and derivation sets 243
6.12 Other derivation sets 245
7 Hadamard designs
7.1 Introduction 249
7.2 Hadamard designs 251
7.3 Equivalent matrices 255
7.4 The codes 258
7.5 Kronecker product constructions 263
7.6 Size 16 and special n-tuples 266
7.7 Geometric constructions 269
7.8 The Paley construction 271
7.9 Bent functions 275
7.10 Regular Hadamard matrices 278
7.11 Hadamard matrices of size 24 284
7.12 Hadamard matrices from Steiner systems 288
8 Steiner systems
8.1 Introduction 295
8.2 Steiner triple and quadruple systems 296
8.3 Unitals 298
8.4 Oval designs 304
8.5 Some Steiner 3-designs 309
8.6 Witt designs and Golay codes 312
Bibliography 317
Glossary 337
Index of Names 339
Index of Terms 344
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