## Perfect Codes and Related Structures

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• Дата: 11-11-2022, 02:18
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Название: Perfect Codes and Related Structures
Автор: Tuvi Etzion
Издательство: World Scientific Publishing
Год: 2022
Страниц: 436
Язык: английский
Формат: pdf (true), epub
Размер: 18.49 MB

In this monograph, we develop the theory of one of the most fascinating topics in coding theory, namely, perfect codes and related structures. Perfect codes are considered to be the most beautiful structure in coding theory, at least from the mathematical side. These codes are the largest ones with their given parameters. The book develops the theory of these codes in various metrics — Hamming, Johnson, Lee, Grassmann, as well as in other spaces and metrics. It also covers other related structures such as diameter perfect codes, quasi-perfect codes, mixed codes, tilings, combinatorial designs, and more. The goal is to give the aspects of all these codes, to derive bounds on their sizes, and present various constructions for these codes. The intention is to offer a different perspective for the area of perfect codes.

For example, in many chapters there is a section devoted to diameter perfect codes. In these codes, anticodes are used instead of balls and these anticodes are related to intersecting families, an area that is part of extremal combinatorics. This is one example that shows how we direct our exposition in this book to both researchers in coding theory and mathematicians interested in combinatorics and extremal combinatorics. New perspectives for MDS codes, different from the classic ones, which lead to new directions of research on these codes are another example of how this book may appeal to both researchers in coding theory and mathematicians.The book can also be used as a textbook, either on basic course in combinatorial coding theory, or as an advance course in combinatorial coding theory.

Information theory, launched by the pioneering work of Shannon, has generated a lot of applications and ten of thousands of research papers. One can easily say, without stretching the truth, that its influence on our daily lives is pervasive. Coding theory, which is one of the important sub-areas of information theory, started with the work of Golay and Hamming. This research area was motivated by engineering problems, and from 1950 until today, with the growth of digital communication, the demand for old and new techniques in coding theory has only increased. Although some basics of the theory are not very difficult, over time more and more sophisticated mathematics has been used and developed in coding theory. This has made the area of coding theory very important to electrical engineers and to computer scientists on one hand and to mathematicians on the other hand. Two of the most important types of codes are error-correcting codes and covering codes.

Contents:

Preface
Introduction
Definitions and Preliminaries
Combinatorial Designs and Bounds
Linear Perfect Codes
Nonlinear Perfect Codes
Density and Quasi-Perfect Codes
Codes with Mixed Alphabets
Binary Constant-Weight Codes
NonBinary Constant-Weight Codes
Codes Over Subspaces
The Lee and the Manhattan Metrics
Tiling with a Cluster of Unit Cubes
Codes in Other Metrics
Bibliography

Readership: Undergraduate and graduate students, researchers in coding theory and mathematicians interested in combinatorics and extremal combinatorics.

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