To counteract, designers of such systems often use error-correcting codes – developed in Part 1 of famous paper by Claude Shannon ”A Mathematical Theory. C.E. Shannon, The Mathematical Theory of Communication, Bell F.J. McWilliams, N.J.A. Sloane, The Theory of Error Correcting Codes. Introduction to the theoryof error-correcting codes. Tsuyoshi Uehara (上原. 健) In this note we introduce the basic theory of error-correcting codes, showing.

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Course in Coding Theory [34]; J. H. van Lint, Introduction to Coding Theory. [40]; and tion theory, and also the branch of it called error-correcting codes. Since. A complete introduction to the many mathematical tools used to solve practical problems in coding. Mathematicians have been fascinated with. It is our goal to present the theory of error-correcting codes in a simple, easily See also the papers on comma-free codes mentioned in the introduction.

Perfect codes Block codes are tied to the sphere packing problem, which has received some attention over the years. In two dimensions, it is easy to visualize. Take a bunch of pennies flat on the table and push them together. The result is a hexagon pattern like a bee's nest. But block codes rely on more dimensions which cannot easily be visualized. The powerful 24,12 Golay code used in deep space communications uses 24 dimensions. If used as a binary code which it usually is the dimensions refer to the length of the codeword as defined above.

Luxemburg, F.

Peterson, I. Singer, and A. MacWilliams N. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. New York. Oxford Sole distributors for the U. Although it has its origins in an engineering problem, the subject has developed by using more and more sophisticated mathematical techniques.

It is our goal to present the theory of error-correcting codes in a simple, easily understandable manner, and yet also to cover all the important aspects of the subject. Thus the reader will find both the simpler families of codes - for example, Hamming, BCH, cyclic and Reed-Muller codes - discussed in some detail, together with encoding and decoding methods, as well as more advanced topics such as quadratic residue, Golay, Goppa, alternant, Kerdock, Preparata, and self-dual codes and association schemes.

Our treatment of bounds on the size of a code is similarly thorough. We discuss both the simpler results - the sphere-packing, Plotkin, Elias and Garshamov bounds - as well as the very powerful linear programming method and the McEliece-Rodemich-Rumsey-Welch bound, the best asymptotic result known.

An appendix gives tables of bounds and of the best codes presently known of length up to Having two authors has helped to keep things simple: by the time we both understand a chapter, it is usually transparent.

Therefore this book can be used both by the beginner and by the expert, as an introductory textbook and as a reference book, and both by the engineer and the mathematician. Of course this has not resulted in a thin book, and so we suggest the following meilus: An elementary first course on coding theory for mathematicians: Ch.

A second course for mathematicians: Ch. An elementary first course on coding theory for engineers: Ch. A second course for engineers: Ch. There is then a lot of rich food left for an advanced course: the rest of Chapters 2, 6, 11 and 14, followed by Chapters 15, 18, 19, 20 and 21 - a feast!

The following are the principal codes discussed: Alternant, Ch. Encoding methods are given for: Linear codes, Ch. I , 02; Cyclic codes, Ch. Decoding methods are given for: Linear codes, Ch. When reading the book, keep in mind this piece of advice, which should be given in every preface: if you get stuck on a section, skip it, but keep reading! Starred sections are difficult or dull, and can be omitted on the first or even second reading.

The book ends with an extensive bibliography. Because coding theory overlaps with so many other subjects computers, digital systems, group theory, number theory, the design of experiments, etc. Unfortunately this means that the usual indexing and reviewing journals are not always helpful. We have therefore felt an obligation to give a fairly comprehensive bibliography. The notes at the ends of the chapters give sources for the theorems, problems and tables, as well as small bibliographies for some of the topics covered or not covered in the chapter.

Only block codes for correcting random errors are discussed; we say little about codes for correcting other kinds of errors bursts or transpositions or about variable length codes, convolutional codes or source codes see the Viii Preface Notes to Ch.

Furthermore we have often considered only binary codes, which makes the theory a lot simpler. Most writers take the opposite point of view: they think in binary but publish their results over arbitrary fields.

There are a few topics which were includeh in the original plan for the book but have been reluctantly omitted for reasons of space: i Gray codes and snake-in-the-box codes-see Adelson et al. See also the remarks on codes for synchronizing in the Notes to Ch.

The following books and monographs on coding theory are our predecessors: Berlekamp [,], Blake and Mullin [], Cameron and Van Lint [], Golomb [], Lin [], Van Lint [], Massey [a], Peterson [a], Peterson and Weldon [], Solomon [] and Sloane [ al; while the following collections contain some of the papers in the bibliography: Berlekamp [], Blake [], the special issues [a, ,, Hartnett [], Mann [] and Slepian [].

See also the bibliography []. We owe a considerable debt to several friends who read the first draft very carefully, made numerous corrections and improvements, and frequently saved us from dreadful blunders. In particular we should like to thank I. Blake, P. Delsarte, J.

Goethals, R. Graham, J.

Longo, C. Mallows, J. McKay, V. Pless, H. Pollak, L. Rudolph, D. Sarwate, many other colleagues at Bell Labs, and especially A.

Odlyzko for Preface ix their help. Not all of their suggestions have been followed, however, and the authors are fully responsible for the remaining errors.

This conventional remark is to be taken seriously. We should also like to thank all the typists at Bell Labs who have helped with the book at various times, our secretary Peggy van Ness who has helped in countless ways, and above all Marion Messersmith who has typed and retyped most of the chapters. Sam Lomonaco has very kindly helped us check the galley proofs.

Preface to the third printing We should like to thank many friends who have pointed out errors and misprints. The corrections have either been made in the text or are listed below. A Russian edition was published in by Svyaz Moscow , and we are extremely grateful to L. Bassalygo, I. Grushko and V. Zinov'ev for producing a very careful translation. They supplied us with an extensive list of corrections. MacWilliams Identities.

Linear Programming bound. Decoding RS codes. Local unambiguous decoding of some Hadamard codes and Reed-Muller codes. References Some standard references for coding theory are listed below.

We won't follow any particular one of these. But the material covered can probably be found in some disguise or other in any of these. Theory and Practice of Error-Control Codes. Richard E. Addison-Wesley, Reading, Massachusetts, The Theory of Error Correcting Codes. MacWilliams and N.

North-Holland, Amsterdam, Introduction to Coding Theory. Jacobus H. Springer-Verlag, Berlin, While 6.