Error-Correcting Codes and Finite Fields

✍ Scribed by Oliver Pretzel

• Publisher: Oxford University Press, USA
• Year: 1992
• Tongue: English
• Leaves: 412
• Series: Oxford Applied Mathematics and Computing Science Series
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Titlepage - Error-Correcting Codes and Finite Fields......Page 2
Preface......Page 6
Contents......Page 8
PART 1 BASIC CODING THEORY......Page 14
1 Introduction......Page 16
2 Block codes, weight, and distance......Page 26
4 Errorprocessing for linear codes......Page 60
5 Hamming codes and the binary Golay codes......Page 76
Appendix LA Linear algebra......Page 92
PART 2 FINITE FIELDS......Page 106
6 Introduction and an example......Page 108
7 Euclid's algorithm......Page 119
8 Invertible and irreducible elements......Page 135
9 The construction of fields......Page 149
10 The structure of finite fields......Page 164
11 Roots of polynomials......Page 179
12 Primitive elements......Page 192
Appendix PF Polynomials over a field......Page 204
PART 3 BCH CODES AND OTHER POLYNOMIAL CODES......Page 212
13 BCH codes as subcodes of Hamming codes......Page 214
14 BCH codes as polynomial codes......Page 229
15 BCH error correction: (1) the fundamental equation......Page 246
16 BCH error correction: (2) an algorithm......Page 262
17 Reed-Solomon codes and burst error correction......Page 280
18 Bounds on codes......Page 300
PART 4 CLASSICAL AND GEOMETRIC GOPPA CODES......Page 314
19 Classical Goppa codes......Page 316
20 Classical Goppa codes: error processing......Page 333
21 Introduction to algebraic curves......Page 346
22 Functions on algebraic curves......Page 356
23 A survey of the theory of algebraic curves......Page 368
24 Geometric Goppa codes......Page 381
25 An error pro.cessor for geometric Goppa codes......Page 392
Bibliography......Page 404
Index......Page 407