Error-Correcting Codes and Finite Fields / Edition 1

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This book provides the reader with all the tools necessary to implement modern error-processing techniques. It assumes only a basic knowledge of linear algebra and develops the mathematical theory in parallel with the codes. Central to the text are worked examples which motivate and explain the theory. The book is in four parts. The first introduces the basic ideas of coding theory. The second and third parts cover the theory of finite fields and give a detailed treatment of BCH and Reed-Solomon codes. These parts are linked by their use of Euclid's algorithm as a central technique. The fourth part is devoted to Goppa codes, both classical and geometric, concluding with the Skorobogatov-Vladut error processor. A special feature of this part is a simplified (but rigorous) treatment of the geometry of curves. The book is intended for the advanced instruction of engineers and computer scientists.
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Meet the Author

Imperial College of Science and Technology, London
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Table of Contents

Pt. 1 Basic Coding Theory
1 Introduction 3
2 Block codes, weight, and distance 13
3 Linear codes 27
4 Error processing for linear codes 47
5 Hamming codes and the binary Golay codes 63
Appendix LA: Linear algebra 79
Pt. 2 Finite Fields
6 Introduction and an example 95
7 Euclid's algorithm 106
8 Invertible and irreducible elements 122
9 The construction of fields 136
10 The structure of finite fields 151
11 Roots of polynomials 166
12 Primitive elements 179
Appendix PF: Polynomials over a field 191
Pt. 3 BCH Codes and Other Polynomial Codes
13 BCH codes as subcodes of Hamming codes 201
14 BCH codes as polynomial codes 216
15 BCH error correction: (1) the fundamental equation 233
16 BCH error correction: (2) an algorithm 249
17 Reed-Solomon codes and burst error correction 267
18 Bounds on codes 287
Pt. 4 Classical and Geometric Goppa Codes
19 Classical Goppa codes 303
20 Classical Goppa codes: error processing 320
21 Introduction to algebraic curves 333
22 Functions on algebraic curves 343
23 A survey of the theory of algebraic curves 355
24 Geometric Goppa codes 368
25 An error processor for geometric Goppa codes 379
Bibliography 391
Index 394
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