Self-dual codes and invariant theory

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.

Preface	p. v
List of Symbols	p. xiv
List of Tables	p. xxv
List of Figures	p. xxvii
1 The Type of a Self-Dual Code	p. 1
1.1 Quadratic maps	p. 2
1.2 Self-dual and isotropic codes	p. 4
1.3 Twisted modules and their representations	p. 5
1.4 Twisted rings and their representations	p. 6
1.5 Triangular twisted rings	p. 9
1.6 Quadratic pairs and their representations	p. 11
1.7 Form rings and their representations	p. 13
1.8 The Type of a code	p. 15
1.9 Triangular form rings	p. 18
1.10 Matrix rings of form rings and their representations	p. 19
1.11 Automorphism groups of codes	p. 22
1.12 Shadows	p. 24
2 Weight Enumerators and Important Types	p. 29
2.1 Weight enumerators of codes	p. 29
2.2 MacWilliams identity and generalizations	p. 35
2.2.1 The weight enumerator of the shadow	p. 39
2.3 Catalogue of important types	p. 39
2.3.1 Binary codes	p. 40
2	p. 40
2I	p. 41
2II	p. 41
2S	p. 41
2.3.2 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
2.3.3 Hermitian codes	p. 47
4H	p. 47
q[superscript H]	p. 47
[Characters not reproducible]	p. 48
2.3.4 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
2.3.5 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
2.3.6 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
2.3.7 Linear codes over p-adic integers	p. 60
Z[subscript p]	p. 60
More general p-adic integers	p. 60
2.4 Examples of self-dual codes	p. 60
2.4.1 2: Binary codes	p. 60
2I: Singly-even binary self-dual codes	p. 61
2II: Doubly-even binary self-dual codes	p. 61
2.4.2 4E: Euclidean self-dual codes over F[subscript 4]	p. 64
2.4.3 q[superscript E] (even or odd): Euclidean self-dual codes over F[subscript q]	p. 65
2.4.4 [Characters not reproducible]: Generalized doubly-even self-dual codes	p. 65
2.4.5 3: Euclidean self-dual codes over F[subscript 3]	p. 67
2.4.6 4H: Hermitian self-dual codes over F[subscript 4]	p. 68
2.4.7 q[superscript H]: Hermitian self-dual linear codes over F[subscript q]	p. 68
2.4.8 4H+: Trace-Hermitian additive codes over F[subscript 4]	p. 69
2.4.9 4Z: Self-dual codes over Z/4Z	p. 70
2.4.10 Codes over other Galois rings	p. 76
2.4.11 Z[subscript p]: Codes over the p-adic numbers	p. 77
2.5 The Gleason-Pierce Theorem	p. 80
3 Closed Codes	p. 83
3.1 Bilinear forms and closed codes	p. 83
3.2 Families of closed codes	p. 86
3.2.1 Codes over commutative rings	p. 88
3.2.2 Codes over quasi-Frobenius rings	p. 89
3.2.3 Algebras over a commutative ring	p. 90
3.2.4 Direct summands	p. 94
3.3 Representations of twisted rings and closed codes	p. 94
3.4 Morita theory	p. 96
3.5 New representations from old	p. 98
3.5.1 Subquotients and quotients	p. 98
3.5.2 Direct sums and products	p. 99
3.5.3 Tensor products	p. 100
4 The Category Quad	p. 103
4.1 The category of quadratic groups	p. 104
4.2 The internal hom-functor IHom	p. 108
4.3 Properties of quadratic rings	p. 113
4.4 Morita theory for quadratic rings	p. 116
4.5 Morita theory for form rings	p. 120
4.6 Witt rings, groups and modules	p. 121
5 The Main Theorems	p. 129
5.1 Parabolic groups	p. 130
5.2 Hyperbolic co-unitary groups	p. 131
5.2.1 Generators for the hyperbolic co-unitary group	p. 136
5.3 Clifford-Weil groups	p. 139
5.4 Scalar elements in C([rho])	p. 142
5.5 Clifford-Weil groups and full weight enumerators	p. 149
5.6 Results from invariant theory	p. 155
5.6.1 Molien series	p. 155
5.6.2 Relative invariants	p. 158
5.6.3 Construction of invariants using differential operators	p. 160
5.6.4 Invariants and designs	p. 161
5.7 Symmetrizations	p. 162
5.8 Example: Hermitian codes over F[subscript 9]	p. 167
6 Real and Complex Clifford Groups	p. 171
6.1 Background	p. 171
6.2 Runge's theorems	p. 174
6.3 The real Clifford group C[subscript m]	p. 177
6.4 The complex Clifford group X[subscript m]	p. 182
6.5 Barnes-Wall lattices	p. 184
6.6 Maximal finiteness in real case	p. 188
6.7 Maximal finiteness in complex case	p. 190
6.8 Automorphism groups of weight enumerators	p. 190
7 Classical Self-Dual Codes	p. 193
7.1 Quasisimple form rings	p. 193
7.2 Split type	p. 195
7.2.1 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
7.3 Hermitian type	p. 201
7.3.1 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
7.4 Orthogonal (or Euclidean) type, p odd	p. 207
7.4.1 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
7.5 Symplectic type, p odd	p. 213
7.5.1 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
7.6 Characteristic 2, orthogonal and symplectic types	p. 215
7.6.1 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
7.6.2 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
7.6.3 [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
7.6.4 [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
8 Further Examples of Self-Dual Codes	p. 227
8.1 m[superscript Z]: Codes over Z/mZ	p. 227
8.2 4Z: Self-dual codes over Z/4Z	p. 230
8.2.1 4Z: Type I self-dual codes over Z/4Z	p. 230
8.2.2 [Characters not reproducible]: Type I self-dual codes over Z/4Z containing 1	p. 231
8.2.3 Same, with 1 in the shadow	p. 233
8.2.4 [Characters not reproducible]: Type II self-dual codes over Z/4Z	p. 233
8.2.5 [Characters not reproducible]: Type II self-dual codes over Z/4Z containing 1	p. 234
8.3 8]: Self-dual codes over Z/8Z	p. 234
8.4 Codes over more general Galois rings	p. 235
8.4.1 GR(p[superscript e], f)[superscript E]: Euclidean self-dual GR(p[superscript e], f)-linear codes	p. 236
8.4.2 GR(p[superscript e], f)[superscript H]: Hermitian self-dual GR(p[superscript e], f)-linear codes	p. 238
8.4.3 GR(p[superscript e], 2l)[superscript H+]: Trace-Hermitian GR(p[superscript e], l)-linear codes	p. 239
8.4.4 Clifford-Weil groups for GR(4, 2)	p. 239
8.5 Self-dual codes over F[subscript q superscript 2] + F [subscript q superscript 2] u	p. 243
9 Lattices	p. 249
9.1 Lattices and theta series	p. 252
9.1.1 Preliminary definitions	p. 252
9.1.2 Modular lattices and Atkin-Lehner involutions	p. 255
9.1.3 Shadows	p. 260
9.1.4 Jacobi forms	p. 261
9.1.5 Siegel theta series	p. 261
Jacobi-Siegel theta series and Riemann theta functions	p. 265
Riemann theta functions with Harmonic coefficients	p. 268
9.1.6 Hilbert theta series	p. 269
9.2 Positive definite form R-algebras	p. 272
9.3 Half-spaces	p. 274
9.4 Form orders and lattices	p. 276
9.5 Even and odd unimodular lattices	p. 278
9.6 Gluing theory for codes	p. 280
9.7 Gluing theory for lattices	p. 282
10 Maximal Isotropic Codes and Lattices	p. 285
10.1 Maximal isotropic codes	p. 286
10.2 Maximal isotropic doubly-even binary codes	p. 290
10.3 Maximal isotropic even binary codes	p. 293
10.4 Maximal isotropic ternary codes	p. 293
10.5 Maximal isotropic additive codes over F[subscript 4]	p. 298
10.6 Maximal isotropic codes over Z/4Z	p. 298
10.7 Maximal even lattices	p. 301
10.7.1 Maximal even lattices of determinant 3k	p. 304
10.7.2 Maximal even and integral lattices of determinant 2k	p. 306
11 Extremal and Optimal Codes	p. 313
11.1 Upper bounds	p. 314
11.1.1 Extremal weight enumerators and the LP bound	p. 314
11.1.2 Self-dual binary codes, 2II and 2I	p. 317
11.1.3 Some other types	p. 321
11.1.4 A new definition of extremality	p. 324
11.1.5 Asymptotic upper bounds	p. 326
11.2 Lower bounds	p. 328
11.3 Tables of extremal self-dual codes	p. 331
11.3.1 Binary codes	p. 331
11.3.2 Type 3: Ternary codes	p. 336
11.3.3 Types 4E and [Characters not reproducible]: Euclidean self-dual codes over F[subscript 4]	p. 338
11.3.4 Type 4H: Hermitian linear self-dual codes over F[subscript 4]	p. 339
11.3.5 Types 4H+ and 4[Characters not reproducible]: Trace-Hermitian codes over F[subscript 4]	p. 340
11.3.6 Type 4Z: Self-dual codes over Z/4Z	p. 342
11.3.7 Other types	p. 345
12 Enumeration of Self-Dual Codes	p. 347
12.1 The mass formulae	p. 347
12.2 Enumeration of binary self-dual codes	p. 350
Interrelations between types 2I and 2II	p. 356
12.3 Type 3: Ternary self-dual codes	p. 360
12.3.1 Types 4E and [Characters not reproducible]: Euclidean self-dual codes over F[subscript 4]	p. 363
12.4 Type 4H: Hermitian self-dual codes over F[subscript 4]	p. 363
12.5 Type 4H+: Trace-Hermitian additive codes over F[subscript 4]	p. 365
12.6 Type 4Z: Self-dual codes over Z/4Z	p. 366
12.7 Other enumerations	p. 367
13 Quantum Codes	p. 369
13.1 Definitions	p. 370
13.2 Additive and symplectic quantum codes	p. 373
13.3 Hamming weight enumerators	p. 376
13.4 Linear programming bounds	p. 381
13.5 Other alphabets	p. 382
13.6 A table of quantum codes	p. 385
References	p. 391
Index	p. 417

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