Selected unsolved problems in coding theory

211 Pages·2011·1.195 MB·English

by  David Joyner, Jon-Lark Kim (auth.)

Applied and Numerical Harmonic Analysis
SeriesEditor
JohnJ.Benedetto
UniversityofMaryland
CollegePark,MD,USA
EditorialAdvisoryBoard
AkramAldroubi JelenaKovacˇevic´
VanderbiltUniversity CarnegieMellonUniversity
Nashville,TN,USA Pittsburgh,PA,USA
AndreaBertozzi GittaKutyniok
UniversityofCalifornia UniversityofOsnabrück
LosAngeles,CA,USA Osnabrück,Germany
DouglasCochran MauroMaggioni
ArizonaStateUniversity DukeUniversity
Phoenix,AZ,USA Durham,NC,USA
HansG.Feichtinger ZuoweiShen
UniversityofVienna NationalUniversityofSingapore
Vienna,Austria Singapore,Singapore
ChristopherHeil ThomasStrohmer
GeorgiaInstituteofTechnology UniversityofCalifornia
Atlanta,GA,USA Davis,CA,USA
StéphaneJaffard YangWang
UniversityofParisXII MichiganStateUniversity
Paris,France EastLansing,MI,USA
David Joyner (cid:2) Jon-Lark Kim
Selected
Unsolved
Problems in
Coding Theory
DavidJoyner Jon-LarkKim
MathematicsDepartment DepartmentofMathematics
USNavalAcademy UniversityofLouisville
ChauvenetHall NaturalSciencesBuilding328
HollowayRoad572C Louisville,KY40292
Annapolis,Maryland21402 USA
USA [e-mail]
[e-mail]
ISBN978-0-8176-8255-2 e-ISBN978-0-8176-8256-9
DOI10.1007/978-0-8176-8256-9
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011935547
Mathematics Subject Classification (2010): 94-02, 94B05, 94B25, 94B27, 11T71, 14G50, 05B05,
05B15
©SpringerScience+BusinessMedia,LLC2011
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein
connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
www.birkhauser-science.com
Preface
Thisbookisintendedforresearchmathematiciansinterestedinunsolvedproblems,
andgraduatestudentsinmathematicsorengineeringwhoareinterestedinthemath-
ematical side of the theory of error-correcting codes. It also may be of interest to
coding-theoristswhosimplywanttoknowhowtouseSAGEtodocertaincompu-
tationswitherror-correctingcodes.
Strongundergraduatesshouldfindmuchinthisbookofsomeinterestaswell.In
termsofclassroomuse,thistextcouldserveasabasisforaspecialtopicscoursein
thetheoryoferror-correctingcodes.Agoodbackgroundinalgebra,especiallylinear
algebra, would be needed from the student. Some sections also require a strong
backgroundinalgebraicgeometryandnumbertheory.
Codingtheoryisthebranchofmathematicsconcernedwithreliablytransmitting
dataacrossnoisychannels.Inmanycases,onecansimplysubdividethedatastream
intoblocksofafixedlength k andthenencodeeachsuchblockwithsomeredun-
dancytoa“codeword”oflongerlengthn,whichisthentransmitted.Withenough
redundancy, the hope is that the receiver can recover the original k data bits. For
example,inthelate1960stoearly1970sNASA’sMariner9tookpictures1ofMars
suchasinFig.1.BlackandwhiteimagessuchasinFig.1weretransmittedthrough
space back to Earth using the so-called Reed–Muller code of length n=32, with
k=6databitsandn−k=26redundancybits.
Inspiteofover60yearsofintensiveworkfromthebestmindsintheworld,there
aremanyinterestingmathematicalquestionswhichremainunsolvedinthetheoryof
error-correctingcodes.Themodestaimofthisbookwillprovidesome“publicity”
forsomeofthosequestions.
A chapter-by-chapter overview follows. We have tried to order the chapters by
theroughlevelofmathematicalsophisticationrequiredfromthereader.
Chapter 1 contains a brief discussion of some basic terms and results on error-
correcting codes. For example, the binary symmetric channel, entropy and uncer-
1ThisimageofMars’OlympusMonswasfoundontheNASAwebsitehttp://marsprogram.jpl.
nasa.gov/MPF/martianchronicle/martianchron2/index.htmlandisinthepublicdomain.Seealso
http://en.wikipedia.org/wiki/Mariner_9.
v
vi Preface
Fig.1 Mars’OlympusMons
takenbyMariner9
tainty, Shannon’s theorem, the Hamming metric, the weight distribution (or spec-
trum)ofacode,decodingbasics,boundsontheparametersofacode(suchasthe
Singletonbound,Manin’stheorem,andtheGilbert–Varshamovasymptoticbound),
and examples of important codes such as the Hamming codes and the quadratic
residuecodes.SAGEexamplesarescatteredthroughouttoemphasizethecomputa-
tionalaspect.
Chapter2isaveryshortchaptersurveyingcertainaspectsofthebeautifultheory
resultingin theintersectionbetweenself-dualcodes, lattices,andinvarianttheory.
Thisisalargefieldwithseveralexcellentbooksandsurveyarticlesalreadywritten.
We introduce weight enumerator polynomials (and the MacWilliams identity), di-
visiblecodesandtheirclassification,invariantsassociatedtothedifferenttypesof
self-dualcodesarisinginthisclassification,andlatticesarisingfromself-dualcodes.
Thechapterendswithadiscussionofthefamousunsolved(atpresent)problemof
theexistenceofaself-dual [72,36,16] binarycode.Again,someSAGEexamples
aregiven.Afewproofsaresketched,butmostresultsarestatedwithonlyreferences
tooriginalproofs.
Chapter3discussessomefascinatingresultsintheintersectionbetweencoding
theory, block designs, group theory, orthogonal arrays, Latin squares, and recre-
ationalmathematics.AfterintroducingHadamardmatrices(andtheHadamardcon-
jecture,withSAGEexamples),oneofthemostremarkableresultsinallofcoding
theoryisdiscussed,theAssmus–Mattsontheorem.Roughlyspeaking,thistheorem
shows a relationship between certain codes and the construction of certain block
designs. Connections with Latin squares and orthogonal arrays are given. The un-
expectedcombinatorialstructure“hidden”incertain“design-theoretic”codesisex-
emplifiedbytheconstructionsinthesectioninvolvingaGolaycodeandthe“kitten”
and“minimog”constructions.Thelastsectionsofthechapterdiscussrecreational
aspectsofthetheory—strategiesforwinninga“mathematicalblackjack”card-game
andhorsetrack-betting.
Preface vii
Chapter 4 explores an intriguing analogy between the Duursma zeta function
(a recently introduced “invariant” object associated to a linear code) and the zeta
functionattachedtoanalgebraiccurveoverafinitefield.Muchremainsunknown
(atthistime)regardingtheDuursmazetafunction,butthischaptersurveysitsknown
properties (mostly with proofs). Several SAGE examples are given; in fact, SAGE
istheonlymathematicssoftwarepackage(atthistime)withcommandstocompute
Duursmazetafunctions.
Chapter5discussestwoveryhardandunsolvedproblems.Thefirstisanontrivial
estimateforthenumberofsolutions(modp)toapolynomialequationy2=f(x),
where f(x) is a polynomial whose degree is “small” compared to the prime p.
(Whenpissmallcomparedtothedegreeoff,thenWeil’sestimategivesagoodes-
timateofthenumberofsolutions.)Thesecondunsolvedproblemisthebestasymp-
toticboundsforabinarylinearcode.Thesurpriseisthatthesetwoseeminglyunre-
latedproblemsareinfact,rathercloselyrelated.Aspectsofthisrelationship,with
someproofsandSAGEexamples,arediscussedindetail.
Finally, Chap. 6 discusses some aspects of algebraic-geometric codes (or AG
codes,forshort).Thesearecodesarisinggenerallyfromalgebraicvarietiesoverfi-
nitefields,thoughthefocushereisonmodularcurves.Thisisarelativelytechnical
chapter,requiringsomefamiliarityofnumbertheory,algebraicgeometry,andmod-
ularformsandalsoofrepresentationtheoryoffinitegroups.Fittingwiththegeneral
themeofthisbook,thischaptermostlyillustrateshowlittleweknowaboutthealge-
braicstructureofAGcodesarisingfrommodularcurves.Aswithmanyotherareas
ofmathematics,itseemsthatthemoreoneknows,themoreonediscovershowlittle
isreallyknown.
Acknowledgements DJ: I thank John Benedetto for the suggestion to write this
bookandallhisencouragementovertheyears.
JLK:IthankProfessorEmeritusVeraPlessofUniversityofIllinoisatChicago
forteachingmetheinsightofcodingtheory.IalsothankmycoauthorDavidJoyner
forhisencouragement.
For Chap. 3, we thank Alex Ryba and Andy Buchanan for helpful comments,
andAnnCasey,whocoauthored(withDJ)amuchearlierandshorterversion.
ForChap.4,wearegratefultoThannWardforthereferenceto[Sl],KojiChinen
formanyinterestingemailsabouthiswork,andtoCaryHuffmanandIwanDuursma
forveryinterestingconversationsonthistopic.
For Chap. 5, we thank Prof. Amin Shokrollahi of the Ecole Polytechnique
FédéraledeLausanneforhelpfuladviceandProf.FelipeVolochoftheUniversity
of Texas for allowing his construction to be included above. Parts of this (such as
Proposition156)canbefoundinthehonorsthesis[C]ofDJ’sformerstudentGreg
Coy,whowasapleasuretoworkwith.
PartofChap.6waswrittenwithSalahoddinShokranianoftheUniversidadede
Brasília (and Amin Shokrollahi’s uncle!). Other parts were adapted from a paper
written with Amy Ksir (of the US Naval Academy). We also thank D. Prasad and
R. Guralnick for enlightening correspondence and in particular for the references
[KP]and[Ja1].
Contents
1 BackgroundonInformationTheoryandCodingTheory . . . . . . . 1
1.1 BinarySymmetricChannel . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Shannon’sTheorem . . . . . . . . . . . . . . . . . . . . . 3
1.2 ASimpleExample . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 TheHammingMetric . . . . . . . . . . . . . . . . . . . . 9
1.4 LinearBlockCodes . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 DecodingBasics . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.2 HammingCodesoverGF(q) . . . . . . . . . . . . . . . . 14
1.5 BoundsontheParametersofaCode . . . . . . . . . . . . . . . . 16
1.5.1 Question:WhatIs“TheBest”Code? . . . . . . . . . . . . 18
1.5.2 TheFakeSingletonBound . . . . . . . . . . . . . . . . . 21
1.6 QuadraticResidueCodesandOtherGroupCodes . . . . . . . . . 22
1.6.1 AutomorphismGroups . . . . . . . . . . . . . . . . . . . 22
1.6.2 CyclicCodes . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6.3 QuadraticResidueCodes . . . . . . . . . . . . . . . . . . 24
2 Self-dualCodes,Lattices,andInvariantTheory . . . . . . . . . . . . 29
2.1 WeightEnumerators . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 DivisibleCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 SomeInvariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 CodesoverOtherFiniteRings . . . . . . . . . . . . . . . . . . . . 38
2.5 LatticesfromCodes . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.1 ConstructionsfromCodes . . . . . . . . . . . . . . . . . . 42
2.5.2 ThetaFunctionofaLattice . . . . . . . . . . . . . . . . . 43
2.6 MoreProblemsRelatedtoaPrizeProblem . . . . . . . . . . . . . 44
3 Kittens,MathematicalBlackjack,andCombinatorialCodes . . . . . 47
3.1 HadamardMatricesandCodes . . . . . . . . . . . . . . . . . . . 47
3.2 Designs,OrthogonalArrays,LatinSquares,andCodes. . . . . . . 51
3.2.1 ExamplesfromGolayCodes . . . . . . . . . . . . . . . . 53
ix
x Contents
3.2.2 Assmus–MattsonTheorem . . . . . . . . . . . . . . . . . 53
3.2.3 OrthogonalArrays,LatinSquaresandCodes . . . . . . . . 56
3.3 Curtis’Kitten,Conway’sMinimog . . . . . . . . . . . . . . . . . 58
3.3.1 TheMINIMOGDescription . . . . . . . . . . . . . . . . . 61
3.3.2 ConstructionoftheExtendedTernaryGolayCode . . . . . 64
3.3.3 The“col/tet”Construction . . . . . . . . . . . . . . . . . . 65
3.3.4 TheKittenLabeling . . . . . . . . . . . . . . . . . . . . . 66
3.4 Playing“MathematicalBlackjack” . . . . . . . . . . . . . . . . . 67
3.5 PlayingtheHorses . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 TheRiemannHypothesisandCodingTheory . . . . . . . . . . . . . 71
4.1 IntroductiontotheRiemannZetaFunction . . . . . . . . . . . . . 72
4.2 IntroductiontotheDuursmaZetaFunction . . . . . . . . . . . . . 73
4.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 VirtualWeightEnumerators . . . . . . . . . . . . . . . . . 74
4.4 TheZetaPolynomial . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.1 FirstDefinition . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.2 SecondDefinition . . . . . . . . . . . . . . . . . . . . . . 83
4.4.3 ThirdDefinition . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.4 AnalogieswithCurves. . . . . . . . . . . . . . . . . . . . 86
4.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5.1 TheFunctionalEquation. . . . . . . . . . . . . . . . . . . 89
4.5.2 PuncturingPreservesP . . . . . . . . . . . . . . . . . . . 91
4.5.3 TheRiemannHypothesis . . . . . . . . . . . . . . . . . . 91
4.6 Self-reciprocalPolynomials . . . . . . . . . . . . . . . . . . . . . 93
4.6.1 “Smoothness”ofRoots . . . . . . . . . . . . . . . . . . . 94
4.6.2 VariationsonaTheoremofEneström–Kakeya . . . . . . . 94
4.6.3 ALiteratureSurvey . . . . . . . . . . . . . . . . . . . . . 95
4.6.4 Duursma’sConjecture . . . . . . . . . . . . . . . . . . . . 103
4.6.5 AConjectureonZerosofCosineTransforms . . . . . . . . 104
4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.7.1 Komichi’sExample . . . . . . . . . . . . . . . . . . . . . 106
4.7.2 TheExtremalCase. . . . . . . . . . . . . . . . . . . . . . 107
4.7.3 “RandomDivisibleCodes” . . . . . . . . . . . . . . . . . 110
4.7.4 AFormallySelf-dual[26,13,6] -code . . . . . . . . . . . 110
2
4.7.5 ExtremalCodesofShortLength . . . . . . . . . . . . . . 111
4.7.6 Non-self-dualExamples . . . . . . . . . . . . . . . . . . . 112
4.8 ChinenZetaFunctions . . . . . . . . . . . . . . . . . . . . . . . . 113
4.8.1 HammingCodes . . . . . . . . . . . . . . . . . . . . . . . 117
4.8.2 GolayCodes . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.8.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 HyperellipticCurvesandQuadraticResidueCodes . . . . . . . . . . 123
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 PointsonHyperellipticCurvesoverFiniteFields . . . . . . . . . . 124
5.3 Non-AbelianGroupCodes . . . . . . . . . . . . . . . . . . . . . 126