Self-dual Codes And Invariant Theory

by Nebe, Gabriele; Rains, Eric M.; Sloane, N. J. A.

• ISBN13: 9783540307297
• ISBN10: 354030729X
• Format: Hardcover
• Copyright: 2006-05-16
• Publisher: Springer-Verlag New York Inc

Summary

One of the most remarkable and beautiful theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes and their connections with invariant theory. In the past 35 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes. This self-contained book develops a new theory which is powerful enough to include all the earlier generalizations. It is also in part an encyclopedia that gives a very extensive list of the different types of self-dual codes and their properties, including tables of the best codes that are presently known. Besides self-dual codes, the book also discusses two closely-related subjects, lattices and modular forms, and quantum error-correcting codes. This book, written by the leading experts in the subject, has no equivalent in the literature and will be of great interest to mathematicians, communication theorists, computer scientists and physicists.

Table of Contents

Preface	p. v
List of Symbols	p. xiv
List of Tables	p. xxv
List of Figures	p. xxvii
The Type of a Self-Dual Code	p. 1
Quadratic maps	p. 2
Self-dual and isotropic codes	p. 4
Twisted modules and their representations	p. 5
Twisted rings and their representations	p. 6
Triangular twisted rings	p. 9
Quadratic pairs and their representations	p. 11
Form rings and their representations	p. 13
The Type of a code	p. 15
Triangular form rings	p. 18
Matrix rings of form rings and their representations	p. 19
Automorphism groups of codes	p. 22
Shadows	p. 24
Weight Enumerators and Important Types	p. 29
Weight enumerators of codes	p. 29
MacWilliams identity and generalizations	p. 35
The weight enumerator of the shadow	p. 39
Catalogue of important types	p. 39
Binary codes	p. 40
2	p. 40
2I	p. 41
2II	p. 41
2S	p. 41
Euclidean codes	p. 42
4E	p. 42
q[superscript E] (even)	p. 43
[Characters not reproducible]	p. 44
3	p. 45
q[superscript E] (odd)	p. 46
[Characters not reproducible] (odd)	p. 46
Hermitian codes	p. 47
4H	p. 47
q[superscript H]	p. 47
[Characters not reproducible]	p. 48
Additive codes	p. 48
4H+	p. 48
4H+ (even)	p. 49
[Characters not reproducible] (even)	p. 49
[Characters not reproducible] (even)	p. 50
[Characters not reproducible] (even)	p. 50
q[superscript H+] (odd)	p. 50
[Characters not reproducible] (odd)	p. 51
Codes over Galois rings Z/mZ	p. 51
4Z	p. 52
m[superscript Z]	p. 53
[Characters not reproducible]	p. 54
[Characters not reproducible]	p. 54
[Characters not reproducible]	p. 55
[Characters not reproducible]	p. 55
Codes over more general Galois rings	p. 55
GR(p[superscript e], f)[superscript E]	p. 55
GR(p[superscript e], f)[Characters not reproducible]	p. 56
GR(p[superscript e], f)[Characters not reproducible]	p. 56
GR(2e, f)[Characters not reproducible]	p. 57
GR(2e, f)[Characters not reproducible]	p. 57
GR(2e, f)[Characters not reproducible]	p. 58
GR(p[superscript e], f)[superscript H]	p. 58
GR(p[superscript e], f)[Characters not reproducible]	p. 58
GR(p[superscript e], f)[superscript H+]	p. 59
GR(p[superscript e], f)[Characters not reproducible]	p. 59
Linear codes over p-adic integers	p. 60
Z[subscript p]	p. 60
More general p-adic integers	p. 60
Examples of self-dual codes	p. 60
2: Binary codes	p. 60
2I: Singly-even binary self-dual codes	p. 61
2II: Doubly-even binary self-dual codes	p. 61
4E: Euclidean self-dual codes over F[subscript 4]	p. 64
q[superscript E] (even or odd): Euclidean self-dual codes over F[subscript q]	p. 65
[Characters not reproducible]: Generalized doubly-even self-dual codes	p. 65
3: Euclidean self-dual codes over F[subscript 3]	p. 67
4H: Hermitian self-dual codes over F[subscript 4]	p. 68
q[superscript H]: Hermitian self-dual linear codes over F[subscript q]	p. 68
4H+: Trace-Hermitian additive codes over F[subscript 4]	p. 69
4Z: Self-dual codes over Z/4Z	p. 70
Codes over other Galois rings	p. 76
Z[subscript p]: Codes over the p-adic numbers	p. 77
The Gleason-Pierce Theorem	p. 80
Closed Codes	p. 83
Bilinear forms and closed codes	p. 83
Families of closed codes	p. 86
Codes over commutative rings	p. 88
Codes over quasi-Frobenius rings	p. 89
Algebras over a commutative ring	p. 90
Direct summands	p. 94
Representations of twisted rings and closed codes	p. 94
Morita theory	p. 96
New representations from old	p. 98
Subquotients and quotients	p. 98
Direct sums and products	p. 99
Tensor products	p. 100
The Category Quad	p. 103
The category of quadratic groups	p. 104
The internal hom-functor IHom	p. 108
Properties of quadratic rings	p. 113
Morita theory for quadratic rings	p. 116
Morita theory for form rings	p. 120
Witt rings, groups and modules	p. 121
The Main Theorems	p. 129
Parabolic groups	p. 130
Hyperbolic co-unitary groups	p. 131
Generators for the hyperbolic co-unitary group	p. 136
Clifford-Weil groups	p. 139
Scalar elements in C([rho])	p. 142
Clifford-Weil groups and full weight enumerators	p. 149
Results from invariant theory	p. 155
Molien series	p. 155
Relative invariants	p. 158
Construction of invariants using differential operators	p. 160
Invariants and designs	p. 161
Symmetrizations	p. 162
Example: Hermitian codes over F[subscript 9]	p. 167
Real and Complex Clifford Groups	p. 171
Background	p. 171
Runge's theorems	p. 174
The real Clifford group C[subscript m]	p. 177
The complex Clifford group X[subscript m]	p. 182
Barnes-Wall lattices	p. 184
Maximal finiteness in real case	p. 188
Maximal finiteness in complex case	p. 190
Automorphism groups of weight enumerators	p. 190
Classical Self-Dual Codes	p. 193
Quasisimple form rings	p. 193
Split type	p. 195
q[superscript lin]: Linear codes over F[subscript q]	p. 196
Clifford-Weil groups	p. 198
F[subscript 2], Genus 1	p. 198
F[subscript 2], Genus 2	p. 199
Hermitian type	p. 201
q[superscript H]: Hermitian self-dual codes over F[subscript q]	p. 202
Clifford-Weil groups	p. 202
The case q = 4	p. 203
The case q = 9	p. 206
Orthogonal (or Euclidean) type, p odd	p. 207
q[superscript E] (odd): Euclidean self-dual codes over F[subscript q]	p. 207
Clifford-Weil groups (q odd)	p. 207
The case q = 3	p. 209
The case q = 3, genus 2	p. 210
The case q = 9	p. 211
The case q = 5	p. 212
Symplectic type, p odd	p. 213
q[superscript H+] (odd): Hermitian F[subscript r]-linear codes over F[subscript q], q = r[superscript 2]	p. 214
Clifford-Weil groups (genus g)	p. 214
The case q = 9, genus 1	p. 215
Characteristic 2, orthogonal and symplectic types	p. 215
q[superscript H+] (even): Hermitian F[subscript r]-linear codes over F[subscript q] q = r[superscript 2]	p. 217
Clifford-Weil groups (genus g)	p. 217
The case q = 4, genus 1	p. 217
The case q = 4, genus 2	p. 219
The case q = 16	p. 220
q[superscript E] (even): Euclidean self-dual F[subscript q]-linear codes	p. 220
Clifford-Weil groups (genus g)	p. 220
The case q = 2	p. 221
The case q = 4	p. 221
[Characters not reproducible] (even): Even Trace-Hermitian F[subscript r]-linear codes	p. 222
Clifford-Weil groups (genus g)	p. 222
The case q = 4, genus 1	p. 223
[Characters not reproducible] (even): Generalized Doubly-even codes over F[subscript q]	p. 224
Clifford-Weil groups (genus g)	p. 224
The case k = F[subscript 2], arbitrary genus	p. 225
The case k - F[subscript 4], genus 1	p. 225
The case k = F[subscript 8]	p. 226
Further Examples of Self-Dual Codes	p. 227
m[superscript Z]: Codes over Z/mZ	p. 227
4Z: Self-dual codes over Z/4Z	p. 230
4Z: Type I self-dual codes over Z/4Z	p. 230
[Characters not reproducible]: Type I self-dual codes over Z/4Z containing 1	p. 231
Same, with 1 in the shadow	p. 233
[Characters not reproducible]: Type II self-dual codes over Z/4Z	p. 233
[Characters not reproducible]: Type II self-dual codes over Z/4Z containing 1	p. 234
8]: Self-dual codes over Z/8Z	p. 234
Codes over more general Galois rings	p. 235
GR(p[superscript e], f)[superscript E]: Euclidean self-dual GR(p[superscript e], f)-linear codes	p. 236
GR(p[superscript e], f)[superscript H]: Hermitian self-dual GR(p[superscript e], f)-linear codes	p. 238
GR(p[superscript e], 2l)[superscript H+]: Trace-Hermitian GR(p[superscript e], l)-linear codes	p. 239
Clifford-Weil groups for GR(4, 2)	p. 239
Self-dual codes over F[subscript q superscript 2] + F [subscript q superscript 2] u	p. 243
Lattices	p. 249
Lattices and theta series	p. 252
Preliminary definitions	p. 252
Modular lattices and Atkin-Lehner involutions	p. 255
Shadows	p. 260
Jacobi forms	p. 261
Siegel theta series	p. 261
Jacobi-Siegel theta series and Riemann theta functions	p. 265
Riemann theta functions with Harmonic coefficients	p. 268
Hilbert theta series	p. 269
Positive definite form R-algebras	p. 272
Half-spaces	p. 274
Form orders and lattices	p. 276
Even and odd unimodular lattices	p. 278
Gluing theory for codes	p. 280
Gluing theory for lattices	p. 282
Maximal Isotropic Codes and Lattices	p. 285
Maximal isotropic codes	p. 286
Maximal isotropic doubly-even binary codes	p. 290
Maximal isotropic even binary codes	p. 293
Maximal isotropic ternary codes	p. 293
Maximal isotropic additive codes over F[subscript 4]	p. 298
Maximal isotropic codes over Z/4Z	p. 298
Maximal even lattices	p. 301
Maximal even lattices of determinant 3k	p. 304
Maximal even and integral lattices of determinant 2k	p. 306
Extremal and Optimal Codes	p. 313
Upper bounds	p. 314
Extremal weight enumerators and the LP bound	p. 314
Self-dual binary codes, 2II and 2I	p. 317
Some other types	p. 321
A new definition of extremality	p. 324
Asymptotic upper bounds	p. 326
Lower bounds	p. 328
Tables of extremal self-dual codes	p. 331
Binary codes	p. 331
Type 3: Ternary codes	p. 336
Types 4E and [Characters not reproducible]: Euclidean self-dual codes over F[subscript 4]	p. 338
Type 4H: Hermitian linear self-dual codes over F[subscript 4]	p. 339
Types 4H+ and 4[Characters not reproducible]: Trace-Hermitian codes over F[subscript 4]	p. 340
Type 4Z: Self-dual codes over Z/4Z	p. 342
Other types	p. 345
Enumeration of Self-Dual Codes	p. 347
The mass formulae	p. 347
Enumeration of binary self-dual codes	p. 350
Interrelations between types 2I and 2II	p. 356
Type 3: Ternary self-dual codes	p. 360
Types 4E and [Characters not reproducible]: Euclidean self-dual codes over F[subscript 4]	p. 363
Type 4H: Hermitian self-dual codes over F[subscript 4]	p. 363
Type 4H+: Trace-Hermitian additive codes over F[subscript 4]	p. 365
Type 4Z: Self-dual codes over Z/4Z	p. 366
Other enumerations	p. 367
Quantum Codes	p. 369
Definitions	p. 370
Additive and symplectic quantum codes	p. 373
Hamming weight enumerators	p. 376
Linear programming bounds	p. 381
Other alphabets	p. 382
A table of quantum codes	p. 385
References	p. 391
Index	p. 417
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