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It is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet Z4. There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on Z4-codes. 本書為英文版。
目次
Preface to the Third Edition Preface to the Second Edition Preface to the First Edition CHAPTER 1 Mathematical Backblound 1.1. Algebra 1.2. Krawtchouk Polynomials 1.3. Combinatorial Theory 1.4. Probability Theory CHAPTER 2 Shannons Theorem 2.1. Introduction 2.2. Shannons Theorem 2.3. On Coding Gain 2.4. Comments 2.5. Problems CHAPTER 3 Linear Codes 3.1. Block Codes 3.2. Linear Codes 3.3. Hamming Codes 3.4. Majority Logic Decoding 3.5. Weight Enumerators 3.6. The Lee Metric 3.7. Comments 3.8. Problems CHAPTER 4 Some Good Codes 4.1. Hadamard Codes and Generalizations 4.2. The Binary Golay Code 4.3. The Temary Golay Code 4.4. Constructing Codes from Other Codes 4.5. Reed-Muller Codes 4.6. Kerdock Codes 4.7. Comments 4.8. Problems CHAPTER 5 Bounds on Codes 5.1. Introduction: The Gilbert Bound 5.2. Upper Bounds 5.3. The Linear Programming Bound 5.4. Comments 5.5. Problems CHAPTER 6 Cyclic Codes 6.1. Definitions 6.2. Generator Matrix and Check Polynomial 6.3. Zeros of a Cyclic Code 6.4. The Idempotent of a Cyclic Code 6.5. Other Representations of Cyclic Codes 6.6. BCH Codes 6.7. Decoding BCH Codes 6.8. Reed-Solomon Codes 6.9. Quadratic Residue Codes 6.10. Binary Cyclic Codes of Length 2n(n odd) 6.11. Generalized Reed-Muller Codes 6.12. Comments 6.13. Problems CHAPTER 7 Perfect Codes and Uniformly Packed Codes 7.1. Lloyds Theorem 7.2. The Characteristic Polynomial of a Codetents 7.3. Uniformly Packed Codes 7.4. Examples of Uniformly Packed Codes 7.5. Nonexistence Theorems 7.6. Comments 7.7. Problems CHAPTER 8 Codes over ■ 8.1. Quaternary Codes 8.2. Binary Codes Derived from Codes over ■ 8.3. Galois Rings over ■ 8.4. Cyclic Codes over■ 8.5. Problems CHAPTER 9 Goppa Codes 9.1. Motivation 9.2. Goppa Codes 9.3. The Minimum Distance of Goppa Codes 9.4. Asymptotic Behaviour of Goppa Codes 9.5. Decoding Goppa Codes 9.6. Generalized BCH Codes 9.7. Comments 9.8. Problems CHAPTER I0 Algebraic Geometry Codes 10.1. Introduction 10.2. Algebraic Curves 10.3. Divisors 10.4. Differentials on a Curve 10.5. The Riemann-Roch Theorem 10.6. Codes from Algebraic Curves 10.7. Some Geometric Codes 10.8. Improvement of the Gilbert-Varshamov Bound 10.9. Comments 10.10. Problems CHAPTER 11 Asymptotically Good Algebraic Codes 11.1. A Simple Nonconstructive Example 11.2. Justesen Codes 11.3. Comments 11.4. Problems CHAPTER 12 Arithmetic Codes 12.1. AN Codes 12.2. The Arithmetic and Modular Weight 12.3. Mandelbaum-Barrows Codes 12.4. Comments 12.5. Problems CHAPTER 13 Convolutional Codes 13.1. Introduction 13.2. Decoding of Convolutional Codes 13.3. An Analog of the Gilbert Bound for Some Convolutional Codes 13.4. Construction of Convolutional Codes from Cyclic Block Codes 13.5. Automorphisms of Convolutional Codes 13.6. Comments 13.7. Problems Hints and Solutions to Problems References Index
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