Lint Introduction to Coding Theory

2. Auflage 1992
ISBN: 978-3-662-00174-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 186 Seiten, eBook

Reihe: Mathematics and Statistics (R0)

The first edition of this book was conceived in 1981 as an alternative to outdated, oversized, or overly specialized textbooks in this area of discrete mathematics-a field that is still growing in importance as the need for mathematicians and computer scientists in industry continues to grow. The body of the book consists of two parts: a rigorous, mathematically oriented first course in coding theory followed by introductions to special topics. The second edition has been largely expanded and revised. The main editions in the second edition are: (1) a long section on the binary Golay code; (2) a section on Kerdock codes; (3) a treatment of the Van Lint-Wilson bound for the minimum distance of cyclic codes; (4) a section on binary cyclic codes of even length; (5) an introduction to algebraic geometry codes. Eindhoven J. H. VAN LINT November 1991 Preface to the First Edition Coding theory is still a young subject. One can safely say that it was born in 1948. It is not surprising that it has not yet become a fixed topic in the curriculum of most universities. On the other hand, it is obvious that discrete mathematics is rapidly growing in importance. The growing need for mathe maticians and computer scientists in industry will lead to an increase in courses offered in the area of discrete mathematics. One of the most suitable and fascinating is, indeed, coding theory.

Zielgruppe

Professional/practitioner

Autoren/Hrsg.

Lint, J.H. van

Inhaltsverzeichnis

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. Comments.- 2.4. 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. Comments.- 3.7. 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 and Algebraic Geometry Codes.- 6.9. Quadratic Residue Codes.- 6.10. Binary Cyclic codes of length 2n (n odd).- 6.11. Comments.- 6.12. 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 Goppa Codes.- 8.1. Motivation.- 8.2. Goppa Codes.- 8.3. The Minimum Distance of Goppa Codes.- 8.4. Asymptotic Behaviour of Goppa Codes.- 8.5. Decoding Goppa Codes.- 8.6. Generalized BCH Codes.- 8.7. Comments.- 8.8. Problems.- 9 Asymptotically Good Algebraic Codes.- 9.1. A Simple Nonconstructive Example.- 9.2. Justesen Codes.- 9.3. Comments.- 9.4. Problems.- 10 Arithmetic Codes.- 10.1. AN Codes.- 10.2. The Arithmetic and Modular Weight.- 10.3. Mandelbaum-Barrows Codes.- 10.4. Comments.- 10.5. Problems.- 11 Convolutional Codes.- 11.1. Introduction.- 11.2. Decoding of Convolutional Codes.- 11.3. An Analog of the Gilbert Bound for Some Convolutional Codes.- 11.4. Construction of Convolutional Codes from Cyclic Block Codes.- 11.5. Automorphisms of Convolutional Codes.- 11.6. Comments.- 11.7. Problems.- Hints and Solutions to Problems.- References.