Fundamentals of Error-Correcting Codes

665 Pages·2010·11.751 MB·English

by  W. Cary Huffman, Vera Pless

Fundamentals of Error-Correcting Codes
FundamentalsofError-CorrectingCodesisanin-depthintroductiontocodingtheoryfrom
bothanengineeringandmathematicalviewpoint.Aswellascoveringclassicaltopics,much
coverage is included of recent techniques that until now could only be found in special-
ist journals and book publications. Numerous exercises and examples and an accessible
writing style make this a lucid and effective introduction to coding theory for advanced
undergraduateandgraduatestudents,researchersandengineers,whetherapproachingthe
subjectfromamathematical,engineering,orcomputersciencebackground.
ProfessorW.CaryHuffman graduatedwithaPhDinmathematicsfromtheCaliforniaInstitute
ofTechnologyin1974.HetaughtatDartmouthCollegeandUnionCollegeuntilhejoined
the Department of Mathematics and Statistics at Loyola in 1978, serving as chair of the
departmentfrom1986through1992.Heisanauthorofapproximately40researchpapers
infinitegrouptheory,combinatorics,andcodingtheory,whichhaveappearedinjournals
suchastheJournalofAlgebra,IEEETransactionsonInformationTheory,andtheJournal
ofCombinatorialTheory.
ProfessorVeraPless was an undergraduate at the University of Chicago and received her
PhD from Northwestern in 1957. After ten years at the Air Force Cambridge Research
Laboratory, she spent a few years at MIT’s project MAC. She joined the University of
Illinois-Chicago’sDepartmentofMathematics,Statistics,andComputerScienceasafull
professorin1975andhasbeenthereeversince.SheisaUniversityofIllinoisScholarand
haspublishedover100papers.

Fundamentals of
Error-Correcting Codes
W. Cary Huffman
LoyolaUniversityofChicago
and
Vera Pless
UniversityofIllinoisatChicago
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 2003
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First published in print format 2003
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ToGayle,Kara,andJonathan
Bill,Virginia,andMike
MinandMary
Thanksforallyourstrengthandencouragement
W.C.H.
TomychildrenNomi,Ben,andDan
fortheirsupport
andgrandchildrenLilah,Evie,andBecky
fortheirlove
V.P.

Contents
Preface pagexiii
1 Basic concepts of linear codes 1
1.1 Threefields 2
1.2 Linearcodes,generatorandparitycheck
matrices 3
1.3 Dualcodes 5
1.4 Weightsanddistances 7
1.5 Newcodesfromold 13
1.5.1 Puncturingcodes 13
1.5.2 Extendingcodes 14
1.5.3 Shorteningcodes 16
1.5.4 Directsums 18
1.5.5 The(u|u+v)construction 18
1.6 Permutationequivalentcodes 19
1.7 Moregeneralequivalenceofcodes 23
1.8 Hammingcodes 29
1.9 TheGolaycodes 31
1.9.1 ThebinaryGolaycodes 31
1.9.2 TheternaryGolaycodes 32
1.10 Reed–Mullercodes 33
1.11 Encoding,decoding,andShannon’sTheorem 36
1.11.1 Encoding 37
1.11.2 DecodingandShannon’sTheorem 39
1.12 SpherePackingBound,coveringradius,and
perfectcodes 48
2 Bounds on the size of codes 53
2.1 A (n,d)and B (n,d) 53
q q
2.2 ThePlotkinUpperBound 58
viii Contents
2.3 TheJohnsonUpperBounds 60
2.3.1 TheRestrictedJohnsonBound 61
2.3.2 TheUnrestrictedJohnsonBound 63
2.3.3 TheJohnsonBoundfor A (n,d) 65
q
2.3.4 TheNordstrom–Robinsoncode 68
2.3.5 Nearlyperfectbinarycodes 69
2.4 TheSingletonUpperBoundandMDScodes 71
2.5 TheEliasUpperBound 72
2.6 TheLinearProgrammingUpperBound 75
2.7 TheGriesmerUpperBound 80
2.8 TheGilbertLowerBound 86
2.9 TheVarshamovLowerBound 87
2.10 Asymptoticbounds 88
2.10.1 AsymptoticSingletonBound 89
2.10.2 AsymptoticPlotkinBound 89
2.10.3 AsymptoticHammingBound 90
2.10.4 AsymptoticEliasBound 92
2.10.5 TheMRRWBounds 93
2.10.6 AsymptoticGilbert–VarshamovBound 94
2.11 Lexicodes 95
3 Finite fields 100
3.1 Introduction 100
3.2 PolynomialsandtheEuclideanAlgorithm 101
3.3 Primitiveelements 104
3.4 Constructingfinitefields 106
3.5 Subfields 110
3.6 Fieldautomorphisms 111
3.7 Cyclotomiccosetsandminimalpolynomials 112
3.8 Traceandsubfieldsubcodes 116
4 Cyclic codes 121
4.1 Factoringxn −1 122
4.2 Basictheoryofcycliccodes 124
4.3 Idempotentsandmultipliers 132
4.4 Zerosofacycliccode 141
4.5 Minimumdistanceofcycliccodes 151
4.6 Meggittdecodingofcycliccodes 158
4.7 Affine-invariantcodes 162
ix Contents
5 BCH and Reed–Solomon codes 168
5.1 BCHcodes 168
5.2 Reed–Solomoncodes 173
5.3 GeneralizedReed–Solomoncodes 175
5.4 DecodingBCHcodes 178
5.4.1 ThePeterson–Gorenstein–ZierlerDecodingAlgorithm 179
5.4.2 TheBerlekamp–MasseyDecodingAlgorithm 186
5.4.3 TheSugiyamaDecodingAlgorithm 190
5.4.4 TheSudan–GuruswamiDecodingAlgorithm 195
5.5 Bursterrors,concatenatedcodes,andinterleaving 200
5.6 Codingforthecompactdisc 203
5.6.1 Encoding 204
5.6.2 Decoding 207
6 Duadic codes 209
6.1 Definitionandbasicproperties 209
6.2 Abitofnumbertheory 217
6.3 Existenceofduadiccodes 220
6.4 Orthogonalityofduadiccodes 222
6.5 Weightsinduadiccodes 229
6.6 Quadraticresiduecodes 237
6.6.1 QRcodesoverfieldsofcharacteristic2 238
6.6.2 QRcodesoverfieldsofcharacteristic3 241
6.6.3 ExtendingQRcodes 245
6.6.4 AutomorphismsofextendedQRcodes 248
7 Weight distributions 252
7.1 TheMacWilliamsequations 252
7.2 Equivalentformulations 255
7.3 Auniquenessresult 259
7.4 MDScodes 262
7.5 Cosetweightdistributions 265
7.6 Weightdistributionsofpuncturedandshortenedcodes 271
7.7 Otherweightenumerators 273
7.8 Constraintsonweights 275
7.9 Weightpreservingtransformations 279
7.10 GeneralizedHammingweights 282