Coding Theory: A First Course

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Cambridge University Press, Feb 12, 2004 - Mathematics - 222 pages
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Preface page xi

1 Introduction 1
Exercises 4

2 Error detection, correction and decoding 5
2.1 Communication channels 5
2.2 Maximum likelihood decoding 8
2.3 Hamming distance 8
2.4 Nearest neighbour/minimum distance decoding 10
2.5 Distance of a code 11
Exercises 14

3 Finite fields 17
3.1 Fields 17
3.2 Polynomial rings 22
3.3 Structure of finite fields 26
3.4 Minimal polynomials 30
Exercises 36

4 Linear codes 39
4.1 Vector spaces over finite fields 39
4.2 Linear codes 45
4.3 Hamming weight 46
4.4 Bases for linear codes 48
4.5 Generator matrix and parity-check matrix 52
4.6 Equivalence of linear codes 56
4.7 Encoding with a linear code 57
4.8 Decoding of linear codes 59




4.8.1 Cosets 59
4.8.2 Nearest neighbour decoding for linear codes 61
4.8.3 Syndrome decoding 62
Exercises 66

5 Bounds in coding theory 75
5.1 The main coding theory problem 75
5.2 Lower bounds 80
5.2.1 Sphere-covering bound 80
5.2.2 Gilbert-Varshamov bound 82
5.3 Hamming bound and perfect codes 83
5.3.1 Binary Hamming codes 84
5.3.2 q-ary Hamming codes 87
5.3.3 Golay codes 88
5.3.4 Some remarks on perfect codes 92
5.4 Singleton bound and MDS codes 92
5.5 Plotkin bound 95
5.6 Nonlinear codes 96
5.6.1 Hadamard matrix codes 98
5.6.2 Nordstrom-Robinson code 98
5.6.3 Preparata codes 99
5.6.4 Kerdock codes 99
5.7 Griesmer bound 100
5.8 Linear programming bound 102
Exercises 106

6 Constructions of linear codes 113
6.1 Propagation rules 113
6.2 Reed-Muller codes 118
6.3 Subfield codes 121
Exercises 126

7 Cyclic codes 133
7.1 Definitions 133
7.2 Generator polynomials 136
7.3 Generator and parity-check matrices 141
7.4 Decoding of cyclic codes 145
7.5 Burst-error-correcting codes 150
Exercises 153



8 Some special cyclic codes 159
8.1 BCH codes 159
8.1.1 Definitions 159
8.1.2 Parameters of BCH codes 161
8.1.3 Decoding of BCH codes 168
8.2 Reed-Solomon codes 171
8.3 Quadratic-residue codes 175
Exercises 183

9 Goppa codes 189
9.1 Generalized Reed-Solomon codes 189
9.2 Altemant codes 192
9.3 Goppa codes 196
9.4 Sudan decoding for generalized RS codes 202
9.4.1 Generation of the (P, k, t)-polynomial 203
9.4.2 Factorization of the (P, k, t)-polynomial 205
Exercises 209

References 215
Bibliography 217
Index 219
 

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Contents

II
1
III
5
V
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VII
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VIII
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IX
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X
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XII
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XLI
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XLII
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LXIX
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