A Course In Error-Correcting Codes

228 Pages·2017·2.387 MB·English

by  Jørn Justesen, Tom Høholdt

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of most interest in research, development, and application. They t
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give a relatively brief presentation of the essential results, g

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emphasizing the interrelations between different methods and o
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proofs of all important results. A sequence of problems at the end e
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of each chapter serves to review the results and give the student an

appreciation of the concepts. In addition, some problems and S
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ISBN 978-3-03719-179-8
www.ems-ph.org
Justesen 2nd | FONT: Rotis, WiesbadenSwing, Times, Helvetica | 4 colors (Euroscala) RB: 21 mm

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Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes
Markus Stroppel, Locally Compact Groups
Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations
Dorothee D. Haroske and Hans Triebel, Distributions, Sobolev Spaces, Elliptic Equations
Thomas Timmermann, An Invitation to Quantum Groups and Duality
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Jørn Justesen
Tom Høholdt
A Course In Error-
Correcting Codes
Second edition
Authors:
Jørn Justesen Tom Høholdt
Department of Photonics Engineering Department of Applied Mathematics and Computer Science
Technical University of Denmark Technical University of Denmark
2800 Kgs. Lyngby 2800 Kgs. Lyngby
Denmark Denmark
E-mail: [email protected] E-mail: [email protected]
2010 Mathematics Subject Classification: 94-01;12-01
Key words: Error-correcting codes, Reed–Solomon codes, convolutional codes, product codes,
graph codes, algebraic geometry codes
First edition published in 2004 by the European Mathematical Society
ISBN 978-3-03719-179-8
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad-
casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use
permission of the copyright owner must be obtained.
© 2017 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
Seminar for Applied Mathematics
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Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany
Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany
∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1
Preface to the second edition
The new edition of the book reflects the fact that both the theory and the appli-
cationshaveevolvedoverthelasttwelveyears. Sometopicshavebeenremoved,
othersexpanded,andnewcodeswereadded.
We want to thank Shorijo Sakata, Knud J. Larsen, and Johan S. Rosenkilde
Nielsenfortheirhelpinimprovingthetext.
Wehavenotincludedreferencestoscientificliterature,industrystandards,or
internet resources. As both theory and applications have diversified, we can no
longerpoint to a few useful starting points for further studies, but the interested
readershouldhavenodifficultyinsearchingforadditionalinformationaboutspe-
cificsubjects.
WashingtonD.C.,Virum,December2016 JørnJustesen,TomHøholdt

Contents
1 BlockCodesforErrorCorrection . . . . . . . . . . . . . . . . . . . 1
1.1 Linearcodesandvectorspaces . . . . . . . . . . . . . . . . . . . 1
1.2 Minimumdistanceandminimumweight . . . . . . . . . . . . . . 4
1.3 SyndromedecodingandtheHammingbound . . . . . . . . . . . 8
1.4 Weightdistributions . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 FiniteFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Fundamentalpropertiesoffinitefields . . . . . . . . . . . . . . . 21
2.2 Polynomialsoverfinitefields . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Reed–SolomoncodesoverF . . . . . . . . . . . . . . . . 26
p
2.3 ThefinitefieldF2m . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 QuadraticpolynomialsoverF2m . . . . . . . . . . . . . . 29
2.4 Minimalpolynomialsandfactorizationofxn(cid:2)1 . . . . . . . . . 30
2.5 Geometriesoverfinitefields . . . . . . . . . . . . . . . . . . . . 35
2.5.1 Affineplanes . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.2 Projectiveplanes . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 CommunicationChannelsandErrorProbability . . . . . . . . . . . 41
3.1 Probabilityandentropy . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Probabilitydistributions . . . . . . . . . . . . . . . . . . . 41
3.1.2 Discretemessagesandentropy . . . . . . . . . . . . . . . 42
3.2 Mutualinformationandcapacityofdiscretechannels . . . . . . . 43
3.2.1 Discretememorylesschannels . . . . . . . . . . . . . . . 43
3.2.2 Approximationsforlongsequences . . . . . . . . . . . . . 46
3.3 Errorprobabilitiesforspecificcodes . . . . . . . . . . . . . . . . 47
3.3.1 Theprobabilityoffailureanderror
forboundeddistancedecoding . . . . . . . . . . . . . . . 47
3.3.2 Boundsformaximumlikelihooddecoding
ofbinaryblockcodes . . . . . . . . . . . . . . . . . . . . 50
3.4 Longcodesandchannelcapacity . . . . . . . . . . . . . . . . . . 53
3.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Reed–SolomonCodesandTheirDecoding . . . . . . . . . . . . . . 61
4.1 Basicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 DecodingReed–Solomoncodes . . . . . . . . . . . . . . . . . . 63
4.3 Alistdecodingalgorithm . . . . . . . . . . . . . . . . . . . . . . 66
viii Contents
4.4 Anotherdecodingalgorithm . . . . . . . . . . . . . . . . . . . . 69
4.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 CyclicCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 Introductiontocycliccodes . . . . . . . . . . . . . . . . . . . . . 77
5.2 Generatorandparitycheckmatricesofcycliccodes . . . . . . . . 79
5.3 CyclicReed–SolomoncodesandBCHcodes . . . . . . . . . . . 80
5.3.1 CyclicReed–Solomoncodes . . . . . . . . . . . . . . . . 81
5.3.2 BCHcodes . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 CycliccodesfromPG.2;F2m/ . . . . . . . . . . . . . . . . . . . 83
5.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1 Definitionsofframesandtheirefficiency . . . . . . . . . . . . . . 93
6.2 Framequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.1 Measuresofquality . . . . . . . . . . . . . . . . . . . . . 95
6.2.2 Paritychecksonframes . . . . . . . . . . . . . . . . . . . 96
6.2.3 Headerprotectioncodes . . . . . . . . . . . . . . . . . . . 97
6.3 Shortblockcodesinframes . . . . . . . . . . . . . . . . . . . . . 97
6.3.1 ReedSolomoncodesandlongBCHcodes . . . . . . . . . 99
6.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 MaximumLikelihoodDecodingandConvolutionalCodes . . . . . . 103
7.1 Definitionsofconvolutionalcodes . . . . . . . . . . . . . . . . . 103
7.2 Codewordsandminimumweights . . . . . . . . . . . . . . . . . 107
7.3 Maximumlikelihooddecoding . . . . . . . . . . . . . . . . . . . 111
7.4 Maximumlikelihooddecodingofblockcodes
andtail-bitingcodes . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.5 Puncturedcodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.6 Correctableerrorpatternsandunitmemorycodes . . . . . . . . . 118
7.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8 CombinationsofSeveralCodes . . . . . . . . . . . . . . . . . . . . . 125
8.1 Productcodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 ProductsofReed–Solomonandbinarycodes
(concatenatedcodes) . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2.1 Parametersofconcatenatedcodes . . . . . . . . . . . . . . 129
8.2.2 Innerconvolutionalcodes . . . . . . . . . . . . . . . . . . 131
8.3 Graphcodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.3.1 Graphsandtheiradjacencymatrices . . . . . . . . . . . . 133
8.3.2 Codesongraphs . . . . . . . . . . . . . . . . . . . . . . . 134
8.3.3 RScodesonplanes . . . . . . . . . . . . . . . . . . . . . 136
8.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Contents ix
9 DecodingReed–SolomonandBCHCodes . . . . . . . . . . . . . . . 141
9.1 Syndromecalculation . . . . . . . . . . . . . . . . . . . . . . . . 141
9.2 TheEuclideanalgorithm . . . . . . . . . . . . . . . . . . . . . . 144
9.3 Reed–SolomonandBCHdecodingwiththeEuclidianalgorithm . 146
9.4 Findingtheerrorpositions . . . . . . . . . . . . . . . . . . . . . 148
9.5 Calculationoferrorvalues . . . . . . . . . . . . . . . . . . . . . 149
9.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10 IterativeDecoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.1 Lowdensityparitycheckcodes . . . . . . . . . . . . . . . . . . . 153
10.2 Bitflipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.2.1 Generalizedsyndromes . . . . . . . . . . . . . . . . . . . 155
10.2.2 Abit-flippingalgorithm . . . . . . . . . . . . . . . . . . . 155
10.2.3 Decodingofprojectivegeometrycodes . . . . . . . . . . . 156
10.3 Decodingbymessagepassing. . . . . . . . . . . . . . . . . . . . 157
10.4 Decodingproductcodesbyiterated(serial)decoding . . . . . . . 162
10.5 Decodingofgraphcodes . . . . . . . . . . . . . . . . . . . . . . 165
10.6 Paralleldecoding . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.6.1 Messagepassingingraphcodes . . . . . . . . . . . . . . . 166
10.6.2 Parallelencodinganddecodingofproductcodes . . . . . . 167
10.6.3 Parallelencodinganddecodingofconvolutionalcodes. . . 168
10.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
11 AlgebraicGeometryCodes . . . . . . . . . . . . . . . . . . . . . . . 173
11.1 Hermitiancodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11.2 DecodingHermitiancodes . . . . . . . . . . . . . . . . . . . . . 178
11.3 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A SomeResultsfromLinearAlgebra . . . . . . . . . . . . . . . . . . . 181
A.1 Vandermondematrices . . . . . . . . . . . . . . . . . . . . . . . 181
A.2 Ausefultheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 184
B CommunicationChannels . . . . . . . . . . . . . . . . . . . . . . . . 185
B.1 Gaussianchannels . . . . . . . . . . . . . . . . . . . . . . . . . . 185
B.2 Gaussianchannelswithquantizedinputandoutput . . . . . . . . 186
B.3 MLDecoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
C Tablesofminimalpolynomials . . . . . . . . . . . . . . . . . . . . . 189
D SolutionstoSelectedProblems . . . . . . . . . . . . . . . . . . . . . 193
D.1 SolutionstoproblemsinChapter1 . . . . . . . . . . . . . . . . . 193
D.2 SolutionstoproblemsinChapter2 . . . . . . . . . . . . . . . . . 196
D.3 SolutionstoproblemsinChapter3 . . . . . . . . . . . . . . . . . 198
D.4 SolutionstoproblemsinChapter4 . . . . . . . . . . . . . . . . . 200