Error-Correcting Codes and Finite Fields [Paperback]

This book provides engineers and computer scientists 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 first part introduces the basic ideas of coding theory. The second and third cover the theory of finite fields and give a detailed treatment of BCH and Reed-Solomon codes. The fourth part is devoted to Goppa codes, both classical and geometric, concluding with the Skorobogatov-Vladut error processor. A special feature is a simplified (but rigorous) treatment of the geometry of curves.

PART I: Basic Coding Theory
1. Introduction
2. Block Codes, Weight, and Distance
3. Linear Codes
4. Error Processing for Linear Codes
5. Hamming Codes and the Binary Golay Codes
PART II: Finite Fields
6. Introduction
7. Euclid's Algorithm
8. Invertible and Irreducible Elements
9. The Construction of Finite Fields
10. The Structure of Finite Fields
11. Roots of Polynomials
12. Primitive Elements
PART III: BCH and Other Cyclic Codes
13. BCH Codes as Subcodes of Hamming Codes
14. BCH Codes as Polynomial Codes
15. Decoding BCH Codes: The Fundamental Equation
16. Decoding BCH Codes: A Decoding Algorithm
17. Reed-Solomon Codes and Burst Error Correction
18. Bounds on Codes
PART IV: Classical and Geometric Goppa Codes
19. Classical Goppa Codes
20. Classical Goppa Codes: Error Processing
21. Introduction to Algebraic Curves
22. Functions on Algebraic Curves
23. A Survey of the Theory of Algebraic Curves
24. Geometric Goppa Codes
25. An Error Processor for Geometric Goppa Codes

Covers the standard course in the theory of finite fields and error-correcting codes, and contains a comprehensive introductionlC"