A First Course in Coding Theory

Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study.

Notation

xi

Introduction to error-correcting codes

1

(10)

The main coding theory problem

11

(20)

An introduction to finite fields

31

(10)

Vector spaces over finite fields

41

(6)

Introduction to linear codes

47

(8)

Encoding and decoding with a linear code

55

(12)

The dual code, the parity-check matrix, and syndrome decoding

67

(14)

The Hamming codes

81

(16)

Perfect codes

97

(16)

Codes and Latin squares

113

(12)

A double-error correcting decimal code and an introduction to BCH codes

125

(16)

Cyclic codes

141

(24)

Weight enumerators

165

(10)

The main linear coding theory problem

175

(16)

MDS codes

191

(10)

Concluding remarks, related topics, and further reading

201

(10)

Solutions to exercises

211

(32)

Bibliography

243

(6)

Index

249

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A First Course in Coding Theory

Summary

Table of Contents

Supplemental Materials