Description

The problems of constructing covering codes and of estimating their parameters are the main concern of this book. It provides a unified account of the most recent theory of covering codes and shows how a number of mathematical and engineering issues are related to covering problems.

Scientists involved in discrete mathematics, combinatorics, computer science, information theory, geometry, algebra or number theory will find the book of particular significance. It is designed both as an introductory textbook for the beginner and as a reference book for the expert mathematician and engineer.

A number of unsolved problems suitable for research projects are also discussed.

Table of Contents

Introduction. Covering problems. Applications. Basic Facts. Codes. The MacWilliams identities. Krawtchouk polynomials. Hamming spheres. Finite fields. Families of error-correcting codes. Designs, constant weight codes, graphs. Constructions. Puncturing and adding a parity check bit. Direct sum. Piecewise constant codes. Variations on the (u, u + v) construction. Matrix construction. Cascading. Optimal short nonbinary codes. Simulated annealing and local search. Normality. Amalgamated direct sum. Normality of binary linear codes. Abnormal binary nonlinear codes. Normality of binary nonlinear codes. Blockwise direct sum. Linear Constructions. Basic facts about linear covering codes. The case R=1; examples of small codes. Saving more than one coordinate. Davydov's basic construction. Lower Bounds. Bounds for the cardinality of the union of K spheres. Balanced codes. Excess bounds for codes with covering radius one. Excess bounds for codes with arbitrary covering radius. The method of linear inequalities. Table on K(n,R). Lower bounds for nonbinary codes. Lower Bounds for Linear Codes. Excess bounds for linear codes. Linear codes with covering radius two and three. Tables for linear codes. Upper Bounds. Codes with given size and distance. Covering radii of subcodes. Covering radius and dual distance. Reed-Muller Codes. Definitions and properties. First order Reed-Muller codes. Reed-Muller codes of order 2 and m—3. Covering radius of Reed-Muller codes of arbitrary order. Algebraic Codes. BCH codes: definitions and properties. 2- and 3-error-correcting BCH codes. Long BCH codes. Normality of BCH codes. Other algebraic codes. Perfect Codes. Perfect linear codes over IFq. A none

Details

No. of pages:
541
Language:
English
Copyright:
© 1997
Published:
Imprint:
North Holland
Electronic ISBN:
9780080530079
Print ISBN:
9780444825117
Print ISBN:
9780444552167

About the authors