Chapter 1.Linear codes 1
1.Linear codes 1
2.Properties of a linear code 5
3.At the receiving end 7
4.More about decoding a linear code 15
5.Error probability 18
6.Shannon’s theorem on the existence of good codes 22
7.Hamming codes 23
8.The dual code 26
9.Construction of new codes from old （Ⅱ） 27
10.Some general properties of a linear code 32
11.Summary of Chapter 1 34
Notes on Chapter 1 34
Chapter 2.Nonlinear codes， Hadamard matrices， designs and the Golay code 38
1.Nonlinear codes 38
2.The Plotkin bound 41
3.Hadamard matrices and Hadamard codes 44
4.Conferences matrices 55
5.t-designs 58
6.An introduction to the binary Golay code 64
7.The Steiner system S（5， 6， 12）， and nonlinear single-error cor-recting codes 70
8.An introduction to the Nordstrom-Robinson code 73
9.Construction of new codes from old （Ⅲ） 76
Notes on Chapter 2 78
Chapter 3.An introduction to BCH codes and finite fields 80
1.Double-error-correcting BCH codes （Ⅰ） 80
2.Construction of the field GF（16） 82
3.Double-error-correcting BCH codes （Ⅱ） 86
4.Computing in a finite field 88
Notes on Chapter 3 92
Chapter 4.Finite fields 93
1.Introduction 93
2.Finite fields： the basic theory 95
3.Minimal polynomials 99
4.How to find irreducible polynomials 107
5.Tables of small fields 109
6.The automorphism group of GF（pm） 112
7.The number of irreducible polynomials 114
8.Bases of GF（pm） over GF（p） 115
9.Linearized polynomials and normal bases 118
Notes on Chapter 4 124
Chapter 5.Dual codes and their weight distribution 125
1.Introduction 125
2.Weight distribution of the dual of a binary linear code 125
3.The group algebra 132
4.Characters 134
5.MacWilliams theorem for nonlinear codes 135
6.Generalized MacWilliams theorems for linear codes 141
7.Properties of Krawtchouk polynomials 150
Notes on Chapter 5 153
Chapter 6.Codes， designs and perfect codes 155
1.Introduction 155
2.Four fundamental parameters of a code 156
3.An explicit formula for the weight and distance distribution 158
4.Designs from codes when s ≤ d’ 160
5.The dual code also gives designs 164
6.Weight distribution of translates of a code 166
7.Designs from nonlinear codes when s’< d 174
8.Perfect codes 175
9.Codes over GF（q） 176
10.There are no more perfect codes 179
Notes on Chapter 6 186
Chapter 7.Cyclic codes 188
1.Introduction 188
2.Definition of a cyclic code 188
3.Generator polynomial 190
4.The check polynomial 194
5.Factors of xn - 1 196
6.t-error-correcting BCH codes 201
7.Using a matrix over GF（qn） to define a code over GF（q） 207
8.Encoding cyclic codes 209
Notes on Chapter 7 214
Chapter 8.Cyclic codes （contd.）： Idempotents and Mattson-Solomon polynomials 216
1.Introduction 216
2.Idempotents 217
3.Minimal ideals， irreducible codes， and primitive idempotents 219
4.Weight distribution of minimal codes 227
5.The automorphism group of a code 229
6.The Mattson-Solomon polynomial 239
7.Some weight distributions 251
Notes on Chapter 8 255
Chapter 9.BCH codes 257
1.Introduction 257
2.The true minimum distance of a BCH code 259
3.The number of information symbols in BCH codes 262
4.A table of BCH codes 266
5.Long BCH codes are bad 269
6.Decoding BCH codes 270
7.Quadratic equations over GF（2m） 277
8.Double-error-correcting BCH codes are quasi-perfect 279
9.The Carlitz-Uchiyama bound 280
10.Some weight distributions are asymptotically normal 282
Notes on Chapter 9 291
Chapter 10.Reed-Solomon and Justesen codes 294
1.Introduction 294
2.Reed-Solomon codes 294
3.Extended RS codes 296
4.Idempotents of RS codes 296
5.Mapping GF（2m） codes into binary codes 298
6.Burst error correction 301
7.Encoding Reed-Solomon codes 301
8.Generalized Reed-Solomon codes 303
9.Redundant residue codes 305
10.Decoding RS codes 306
11.Justesen codes and concatenated codes 306
Notes on Chapter 10 315
Chapter 11.MDS codes 317
1.Introduction 317
2.Generator and parity check matrices 318
3.The weight distribution of an MDS code 319
4.Matrices with every square submatrix nonsingular 321
5.MDS codes from RS codes 323
6.n-arcs 326
7.The kncwn results 327
8.Orthogonal arrays 328
Notes on Chapter 11 329
Chapter 12.Alternant， Goppa and other generalized BCH codes 332
1.Introduction 332
2.Alternant codes 333
3.Goppa codes 338
4.Further properties of Goppa codes 346
5.Extended double-error-correcting Goppa codes are cyclic 350
6.Generalized Srivastava codes 357
7.Chien-Choy generalized BCH codes 360
8.The Euclidean algorithm 362
9.Decoding alternant codes 365
Notes on Chapter 12 368
Chapter 13.Reed-Muller codes 370
Chapter 14.First-order Reed-Muller codes 406
Chapter 15.Second-order Reed-Muller， Kerdock and Preparata codes 433
Chapter 16.Quadratic-residue codes 480
Chapter 17.Bounds on the size of a code 523
Chapter 18.Methods for combining codes 567
Chapter 19.Self-dual codes and invariant theory 596
Chapter 20.The Golay codes 634
Chapter 21.Association schemes 651
Appendix A.Tables of the best codes known 673
Appendix B.Finite geometries 692
Bibliography 703
Index 757