CHAPTER 1 Mathematical Background 1
1.1．Algebra 1
1.2．Krawtchouk Polynomials 14
1.3．Combinatorial Theory 17
1.4．Probability Theory 19
CHAPTER 2 Shannon's Theorem 22
2.1．Introduction 22
2.2．Shannon's Theorem 27
2.3．On Coding Gain 29
2.4．Comments 31
2.5．Problems 32
CHAPTER 3 Linear Codes 33
3.1．Block Codes 33
3.2．Linear Codes 35
3.3．Hamming Codes 38
3.4．Majority Logic Decoding 39
3.5．Weight Enumerators 40
3.6．The Lee Metric 42
3.7．Comments 44
3.8．Problems 45
CHAPTER 4 Some Good Codes 47
4.1．Hadamard Codes and Generalizations 47
4.2．The Binary Golay Code 48
4.3．The Ternary Golay Code 51
4.4．Constructing Codes from Other Codes 51
4.5．Reed-Muller Codes 54
4.6．Kerdock Codes 60
4.7．Comments 61
4.8．Problems 62
CHAPTER 5 Bounds on Codes 64
5.1．Introduction：The Gilbert Bound 64
5.2．Upper Bounds 67
5.3．The Linear Programming Bound 74
5.4．Comments 78
5.5．Problems 79
CHAPTER 6 Cyclic Codes 81
6.1．Definitions 81
6.2．Generator Matrix and Check Polynomial 83
6.3．Zeros of a Cyclic Code 84
6.4．The Idempotent of a Cyclic Code 86
6.5．Other Representations of Cyclic Codes 89
6.6．BCH Codes 91
6.7．Decoding BCH Codes 98
6.8．Reed-Solomon Codes 99
6.9．Quadratic Residue Codes 103
6.10．Binary Cyclic Codes of Length 2n（n odd） 106
6.11．Generalized Reed-Muller Codes 108
6.12．Comments 110
6.13．Problems 111
CHAPTER 7 Perfect Codes and Uniformly Packed Codes 112
7.1．Lloyd's Theorem 112
7.2．The Characteristic Polynomial of a Code 115
7.3．Uniformly Packed Codes 118
7.4．Examples of Uniformly Packed Codes 120
7.5．Nonexistence Theorems 123
7.6．Comments 127
7.7．Problems 127
CHAPTER 8 Codes over Z4 128
8.1．Quaternary Codes 128
8.2．Binary Codes Derived from Codes over Z4 129
8.3．Galois Rings over Z4 132
8.4．Cyclic Codes over Z4 136
8.5．Problems 138
CHAPTER 9 Goppa Codes 139
9.1．Motivation 139
9.2．Goppa Codes 140
9.3．The Minimum Distance of Goppa Codes 142
9.4．Asymptotic Behaviour of Goppa Codes 143
9.5．Decoding Goppa Codes 144
9.6．Generalized BCH Codes 145
9.7．Comments 146
9.8．Problems 147
CHAPTER 10 Algebraic Geometry Codes 148
10.1．Introduction 148
10.2．Algebraic Curves 149
10.3．Divisors 155
10.4．Differentials on a Curve 156
10.5．The Riemann-Roch Theorem 158
10.6．Codes from Algebraic Curves 160
10.7．Some Geometric Codes 162
10.8．Improvement of the Gilbert-Varshamov Bound 165
10.9．Comments 165
10.10．Problems 166
CHAPTER 11 Asymptotically Good Algebraic Codes 167
11.1．A Simple Nonconstructive Example 167
11.2．Justesen Codes 168
11.3．Comments 172
11.4．Problems 172
CHAPTER 12 Arithmetic Codes 173
12.1．AN Codes 173
12.2．The Arithmetic and Modular Weight 175
12.3．Mandelbaum-Barrows Codes 179
12.4．Comments 180
12.5．Problems 180
CHAPTER 13 Convolutional Codes 181
13.1．Introduction 181
13.2．Decoding of Convolutional Codes 185
13.3．An Analog of the Gilbert Bound for Some Convolutional Codes 187
13.4．Construction of Convolutional Codes from Cyclic Block Codes 188
13.5．Automorphisms of Convolutional Codes 191
13.6．Comments 193
13.7．Problems 194
Hints and Solutions to Problems 195
References 218
Index 223