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Fundamentals of error-correcting codes PDF

665 Pages·2003·3.401 MB·English
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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 www.cambridge.org Information on this title: www.cambridge.org/9780521782807 © Cambridge University Press 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. 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

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