Introduction to Coding Theory / by J.H. van Lint.

Format:

Book

Author/Creator:

Lint, Jacobus Hendricus van, 1932- Author.

Series:

Graduate Texts in Mathematics, 2197-5612 ; 86

Language:

English

Subjects (All):

Discrete mathematics.

Geometry, Algebraic.

Number theory.

Discrete Mathematics.

Algebraic Geometry.

Number Theory.

Local Subjects:

Discrete Mathematics.

Algebraic Geometry.

Number Theory.

Physical Description:

1 online resource (XIV, 234 p.)

Edition:

3rd ed. 1999.

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1999.

Language Note:

English

Summary:

It is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4• There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l. -codes. 4 Furthermore, some material has been added that appeared in my Springer Lec­ ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2,a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added. For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10.

Contents:

1 Mathematical Background

1.1. Algebra

1.2. Krawtchouk Polynomials

1.3. Combinatorial Theory

1.4. Probability Theory

2 Shannon’s Theorem

2.1. Introduction

2.2. Shannon’s Theorem

2.3. On Coding Gain

2.4. Comments

2.5. Problems

3 Linear Codes

3.1. Block Codes

3.2. Linear Codes

3.3. Hamming Codes

3.4. Majority Logic Decoding

3.5. Weight Enumerators

3.6. The Lee Metric

3.7. Comments

3.8. Problems

4 Some Good Codes

4.1. Hadamard Codes and Generalizations

4.2. The Binary Golay Code

4.3. The Ternary Golay Code

4.4. Constructing Codes from Other Codes

4.5. Reed—Muller Codes

4.6. Kerdock Codes

4.7. Comments

4.8. Problems

5 Bounds on Codes

5.1. Introduction: The Gilbert Bound

5.2. Upper Bounds

5.3. The Linear Programming Bound

5.4. Comments

5.5. Problems

6 Cyclic Codes

6.1. Definitions

6.2. Generator Matrix and Check Polynomial

6.3. Zeros of a Cyclic Code

6.4. The Idempotent of a Cyclic Code

6.5. Other Representations of Cyclic Codes

6.6. BCH Codes

6.7. Decoding BCH Codes

6.8. Reed—Solomon Codes

6.9. Quadratic Residue Codes

6.10. Binary Cyclic Codes of Length 2n(n odd)

6.11. Generalized Reed—Muller Codes

6.12. Comments

6.13. Problems

7 Perfect Codes and Uniformly Packed Codes

7.1. Lloyd’s Theorem

7.2. The Characteristic Polynomial of a Code

7.3. Uniformly Packed Codes

7.4. Examples of Uniformly Packed Codes

7.5. Nonexistence Theorems

7.6. Comments

7.7. Problems

8 Codes over ?4

8.1. Quaternary Codes

8.2. Binary Codes Derived from Codes over ?4

8.3. Galois Rings over ?4

8.4. Cyclic Codes over ?4

8.5. Problems

9 Goppa Codes

9.1. Motivation

9.2. Goppa Codes

9.3. The Minimum Distance of Goppa Codes

9.4. Asymptotic Behaviour of Goppa Codes

9.5. Decoding Goppa Codes

9.6. Generalized BCH Codes

9.7. Comments

9.8. Problems

10 Algebraic Geometry Codes

10.1. Introduction

10.2. Algebraic Curves

10.3. Divisors

10.4. Differentials on a Curve

10.5. The Riemann—Roch Theorem

10.6. Codes from Algebraic Curves

10.7. Some Geometric Codes

10.8. Improvement of the Gilbert—Varshamov Bound

10.9. Comments

10.10.Problems

11 Asymptotically Good Algebraic Codes

11.1. A Simple Nonconstructive Example

11.2. Justesen Codes

11.3. Comments

11.4. Problems

12 Arithmetic Codes

12.1. AN Codes

12.2. The Arithmetic and Modular Weight

12.3. Mandelbaum—Barrows Codes

12.4. Comments

12.5. Problems

13 Convolutional Codes

13.1. Introduction

13.2. Decoding of Convolutional Codes

13.3. An Analog of the Gilbert Bound for Some Convolutional Codes

13.4. Construction of Convolutional Codes from Cyclic Block Codes

13.5. Automorphisms of Convolutional Codes

13.6. Comments

13.7. Problems

Hints and Solutions to Problems

References.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

ISBN:

3-540-64133-5

3-642-58575-2