HBA Texts in Mathematics

This book is written as a text for a course aimed at 3rd or 4th year students. Only some familiarity with elementary linear algebra and probability is directly assumed, but some maturity is required. The students may specialize in discrete mathematics, computer science, or communication engineering. The book is also a suitable introduction to coding theory for researchers from related fields or for professionals who want to supplement their theoretical basis. The book gives the coding basics for working on projects in any of the above areas, but material specific to one of these fields has not been included. The chapters cover the codes and decoding methods that are currently of most interest in research, development, and application. They give a relatively brief presentation of the essential results, emphasizing the interrelations between different methods and proofs of all important results. A sequence of problems at the end of each chapter serves to review the results and give the student an appreciation of the concepts. In addition, some problems and suggestions for projects indicate direction for further work. The presentation encourages the use of programming tools for studying codes, implementing decoding methods, and simulating performance.
1 Block codes for error-correction
2 Finite fields
3 Bounds on error probability for error-correcting codes
4 Communication channels and information theory
5 Reed-Solomon codes and their decoding
6 Cyclic codes
7 Frames
8 Convolutional codes
9 Maximum likelihood decoding of convolutional codes
10 Combinations of several codes
11 Decoding Reed-Solomon and BCH-codes with the Euclidian algorithm
12 List decoding of Reed-Solomon codes
13 Iterative decoding
14 Algebraic geometry codes
A. Communication channels
B. Solutions to selected problems
C. Table of minimal polynomials
Bibliography
Index