Pretzel O. Error-Correcting Codes and Finite Fields

Clarendon Press, 1992. — 205 p. — (Oxford Applied Mathematics and Computing Science Series).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 book is in four parts. 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. These parts are linked by their uses of Eulid'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 is a simplified (but rigorous) treatment of the geometry of curves.Basic Coding TheoryBlock codes, weight, and distance
Linear codes
Error processing for linear codes
Hamming codes and the binary Golay codesFinite FieldsIntroduction and an example
Euclid's algorithm
Invertible and irreducible elements
The construction of fields
The structure of finite fields
Roots of polynomials
Primitive elementsBCH Codes and other Polynomial CodesBCH codes as subcodes of Hamming codes
BCH codes as polynomial codes
BCH error correction: (1) the fundamental equation
BCH error correction: (2) an algorithm
Reed-Solomon codes and burst error correction
Bounds on codesClassical and Geometric Goppa CodesClassical Goppa codes
Classical Goppa codes: error processing
Introduction to algebraic curves
Functions on algebraic curves
A survey of the theory of algebraic curves
Geometric Goppa codes
An error processor for geometric Goppa codes