Perfect Codes and Related Structures

Perfect Codes and Related Structures

Tuvi Etzion


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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.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.Anticodes;Coding Theory;Combinatorial Designs;Constant-Weight Codes;Diameter Perfect Codes;Lee Codes;Perfect Codes;Quasi-Perfect Codes;Tilings0Key Features:
  • The book is the only one which shows how rich is the theory of perfect codes
  • The book combines topics which are not covered together in any book and also present some topics (diameter perfect codes, mixed codes, subspace codes, etc.) which are not covered by any other book
  • The proofs of the results are very detailed compared to most books
  • The book is mostly self-contained and references are quoted only in the last section of each chapter