Error-Correcting Linear Codes
Classification by Isometry and Applications
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Beschreibung
This text offers a thorough introduction to the mathematical concepts behind the theory of error-correcting linear codes. Care is taken to introduce the necessary algebraic concepts, for instance the theory of finite fields, the polynomial rings over such fields and the ubiquitous concept of group actions that allows the classification of codes by isometry. The book provides in-depth coverage of important topics like cyclic codes and the coding theory used in compact disc players.
The final four chapters cover advanced and algorithmic topics like the classification of linear codes by isometry, the enumeration of isometry classes, random generation of codes, the use of lattice basis reduction to compute minimum distances, the explicit construction of codes with given parameters, as well as the systematic evaluation of representatives of all isometry classes of codes. Up until now, these advanced topics have only been covered in research papers.
The present book provides access to these results at a level which is suitable for graduate students of mathematics, computer science and engineering as well as for researchers.
The fascinating theory of error-correcting codes is a rather new addition to the list of mathematical disciplines. It grew out of the need to communicate information electronically, and is currently no more than 60 years old. - ing an applied discipline by de?nition, a surprisingly large number of pure mathematical areas tie into Coding Theory. If one were to name just the most important connections, one would start of course with Linear Algebra, then list Algebra and Combinatorics, and further mention Number Theory and - ometry as well as Algebraic Geometry. Being a thorough introduction to the ?eld, this book starts from the very beginning, which is the channel model of communication in the presence of noise. From there, we develop the fundamental concepts of error-correcting codes, like the Hamming metric and the maximum likelihood decoding pr- ciple. After discussing dual codes and simple decoding procedures, this book takes an unusual turn. The standard approach would be to move on from there and introduce either more theory or present standard constructions of codes. The approach taken here is different.
Well-written, applications-oriented textbook on coding theory based on earlier succesful German edition
This text offers an introduction to error-correcting linear codes. The book differs from other standard texts in its emphasis on the classification of codes by means of isometry classes. It is based on the successful German edition. The relevant algebraic concepts like finite fields and group actions are developed rigorously. Cyclic codes are discussed in great detail. In the last four chapters these isometry classes are enumerated, and representatives are constructed algorithmically with or without a prescribed automorphism group.
Autor*in
Anton Betten
Themen in »Error-Correcting Linear Codes«
Algebra Computer algorithms coding theory computer science error-correcting codes finite field linear codes combinatorics
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From the reviews:"The theory of error-correcting codes is a … new addition to the list of mathematical disciplines. … This book contains 51 figures and 102 tables. … The book provides access to all results at a level which is proper for graduate students of mathematics and computer science as well as for researchers." (Zlatko Varbanov, Zentralblatt MATH, Vol. 1102 (4), 2007)"This is a thorough treatment of the theory of error-correcting codes. … This book is remarkable because of the enormous amount of material presented (in a very lucid style), but also because of the great variety of mathematical disciplines used … . A beautiful book on applied mathematics!" (H. Mitsch, Monatshefte für Mathematik, Vol. 151 (3), 2007)"The main object of the book under review is an error-correcting linear code. … a motivated reader can profit much from studying this monograph, which contains rich material in one of the rapidly developing areas. … The presentation of material is reader-friendly, arguments are clear and concise, numerous exercises are original and stimulating … . To sum up, the book under review can be strongly recommended to anyone interested in the topic." (Boris È. Kunyavskii, Mathematical Reviews, Issue 2008 h)
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