Codes for Error Detection (Series on Coding Theory and Cryptology)

214 Pages·2007·1.091 MB·English

by  Torleiv Klove

CODES FOR
ERROR DETECTION
Series on Coding Theory and Cryptology
Editors: Harald Niederreiter (National University of Singapore, Singapore) and
San Ling (Nanyang Technological University, Singapore)
Published
Vol. 1 Basics of Contemporary Cryptography for IT Practitioners
B. Ryabko and A. Fionov
Vol. 2 Codes for Error Detection
by T. Kløve
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Series on Coding Theory and Cryptology – Vol. 2
CODES FOR
ERROR DETECTION
Torleiv Kløve
University of Bergen, Norway
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Series on Coding Theory and Cryptology — Vol. 2
CODES FOR ERROR DETECTION
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-270-586-0
ISBN-10 981-270-586-4
Printed in Singapore.
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Preface
There are two basic methods of error control for communication, both in-
volving coding of the messages. The di(cid:11)erences lay in the way the codes
are utilized. The codes used are block codes, which are the ones treated
in this book, or convolutionalcodes. Often the block codes used are linear
codes. With forwarderrorcorrection,the codesareusedtodetectandcor-
recterrors. Inarepeatrequestsystem, the codesareusedto detecterrors,
and, if there are errors,request a retransmission.
Usually it is a much more complex task to correct errors than merely
detect them. Detecting errorshas the same complexity as encoding, which
is usually linear in the length of the codewords. Optimal error correcting
decoding is in general an NP-hard problem, and e(cid:14)cient decoding algo-
rithms are known only for some classes of codes. This has generated much
researchinto (cid:12)nding new classes of codes with e(cid:14)cient decoding as well as
new decoding algorithms for known classes of codes.
There are a number of books on error control, some are listed in the
bibliography at the end of this book. The main theme of almost all these
books is error correcting codes. Error detection tends to be looked upon
as trivial and is covered in a few pages at most. What is then the reason
behindthefollowingbookwhichistotallydevotedtoerrordetectingcodes?
The reasonis, on the one hand, that errordetection combined with repeat
requests is a widely used method of error control, and on the other hand,
that the analysis of the reliability of the information transmission system
with error detection is usually quite complex. Moreover, the methods of
analysisareoftennotsu(cid:14)cientlyknown,andsimplerulesofthe thumbare
used instead.
The main parameter of a code for error detection is its probability of
error detection, and this is the main theme of this book. There are many
vii
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viii Codes for Error Detection
papers devoted to the study of the probability of undetected error, both
forsymmetricchannelsandotherchannels,withorwithoutmemory. They
are spread over many journals and conference proceedings. In Kl(cid:28)ve and
Korzhik(1995),which was published twelve yearsago,we collected the re-
sultsthenknown,mainlyforlinearcodesonthebinarysymmetricchannel,
and presented them in a uni(cid:12)ed form. We also included a number of new
results. In the last twelve years a number of signi(cid:12)cant new results have
beenpublished(inapproximatelyonehundrednewpapers). Inthepresent
book, we have included all the important new results, and we also include
some new unpublished results. As far as possible, the results are given
for both linear and non-linear codes, and for general alphabet size. Many
results previously published for binary codes or for linear codes (or both)
are generalized.
Wehavemainlyrestrictedourselvestochannelswithoutmemoryandto
codesforerrordetectiononly(notcombinederrorcorrectionanddetection
since these results belong mainly with error correcting codes; the topic is
brie(cid:13)y mentioned, however).
Chapter 1 is a short introduction to coding theory, concentrating on
topics that are relevant for error detection. In particular, we give a more
detailed presentation of the distance distribution of codes than what is
common in books on error correction.
Chapter2isthelargestchapter,anditcontainsadetailedaccountofthe
knownresultsontheprobabilityofundetectederrorontheq-arysymmetric
channel. Combineddetectionandcorrectionwillbebrie(cid:13)ymentionedfrom
the error detection point of view.
Chapter3 presents results that areparticularfor the binary symmetric
channel.
Chapter 4 considers codes for some other channels.
Each chapter includes a list of comments and references.
Finally, we givea bibliographyof paperson errordetection and related
topics.
The required background for the reader of this book will be some ba-
sic knowledge of coding theory and some basic mathematical knowledge:
algebra (matrices, groups, (cid:12)nite (cid:12)elds, vector spaces, polynomials) and el-
ementary probability theory.
Bergen, January 2007
T. Kl(cid:28)ve
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Contents
Preface vii
1. Basics on error control 1
1.1 ABC on codes. . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Basic notations and terminology . . . . . . . . . . 1
1.1.2 Hamming weight and distance . . . . . . . . . . . 2
1.1.3 Support of a set of vectors . . . . . . . . . . . . . 2
1.1.4 Extending vectors . . . . . . . . . . . . . . . . . . 3
1.1.5 Ordering . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.6 Entropy. . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.7 Systematic codes . . . . . . . . . . . . . . . . . . 3
1.1.8 Equivalent codes . . . . . . . . . . . . . . . . . . . 4
1.1.9 New codes from old . . . . . . . . . . . . . . . . . 4
1.1.10 Cyclic codes . . . . . . . . . . . . . . . . . . . . . 5
1.2 Linear codes. . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Generator and check matrices for linear codes . . 6
1.2.2 The simplex codes and the Hamming codes . . . . 6
1.2.3 Equivalent and systematic linear codes . . . . . . 7
1.2.4 New linear codes from old . . . . . . . . . . . . . 7
1.2.5 Cyclic linear and shortened cyclic linear codes . . 10
1.3 Distance distribution of codes . . . . . . . . . . . . . . . . 13
1.3.1 De(cid:12)nition of distance distribution . . . . . . . . . 13
1.3.2 The MacWilliams transform . . . . . . . . . . . . 13
1.3.3 Binomial moment . . . . . . . . . . . . . . . . . . 16
1.3.4 Distance distribution of complementary codes . . 20
1.4 Weight distribution of linear codes . . . . . . . . . . . . . 22
1.4.1 Weight distribution . . . . . . . . . . . . . . . . . 22
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