Now we turn to list decoding arbitrary received words with a bounded distance from the ReedSolomon code using Sudans algorithm This algorithm decodes close to a fraction 1 of errors for low rates Then we will see an improvement by Guruswami and Su ID: 69743
Download Pdf The PPT/PDF document "Introduction to Coding Theory CMU Spring..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Lemma8Givenadegree-kpolynomialf(x)2F[X]withf( i)=yifortdi erentpoints( i;yi)2F2:Q( i;yi)=0,if(1;k)-weighteddegreeofQist,theny�f(x)Q(x;y).Proof:Asusual,letR(x):=Q(x;f(x))andnotethatR(x)hastdi erentroots.NowdegreeofRisatmostmaxij:qij6=0i+degx(f)jmaxij:qij6=0i+kjt.HereqijisthecoecientofmonomialxiyjofQ(x;y).Hencey�f(x)Q(x;y).Puttingthesetogether,weobtain:Theorem9(6)Givenparametersn,t,kandnpairs( i;yi)2F2,ift&#x-278;p 2kn,wecanndalldegreekpolynomialsf2F[X]suchthatif( i)=yi tintimepolynomialinn,kandjFj.Moreoverthereareatmostp 2n=kmanysuchf's.Proof:LetD t�1.Then(D+1)(D+2) 2k=t2+t 2k�2nk+1 2k�n.ByLemma??,aQwith(1;k)-weighteddegreet�1exists.ByLemma8,weknowthatiffhast-agreement,y�f(x)Q(x;y).Assumingthatwecanndsuchfactorsinpolynomialtime,wearedone.1.1.3FindingfactorsofQ(X;Y)oftheformY�f(X)Wewilldescribeasimplerandomizedalgorithmgivenin[1,Section4.2].WearegivenabivariatepolynomialQ(X;Y)2F[X;Y]withq=jFjandwewanttondfactorsy�f(x)withdegx(f)k.Ifwecanndanalgorithmthateitherndssuchpolynomialf(x)orconcludesthatnoneexists,wearedone.Wecanrunthisalgorithm,andifitndsf(x),wecanrecurseonthequotientQ(x;y)=(y�f(x)).LetE(X)2F[X]beanirreduciblepolynomialwithdegreek+1.AnexplicitchoiceisE(x)=xq�1� where isageneratorofF.AswehaveseeninHomework2,thisisirreducible.GivensuchE(x),wecanviewQ(x;y)modE(x)asapolynomial,eQ(Y)2eF[Y]withcoecientsintheextensioneldeF=F[X]=(E(X))ofdegreeq�1overF.TheproblemreducestondingrootsofeQ(Y)ineF,whichcanbedoneusingBerlekamp'salgorithmintimepolynomialindegeQandq.2MethodofMultiplicitiesInthissection,wewillremovethefactorp 2andobtainalistdecodingalgorithmwitht�p knduetoGuruswamiandSudan[4].Firstwewillseeanexampleshowingthatwecan'thopetoimprovetheaboveparameters.Example2Considerthefollowingpointcongurationofn=10points.Fork=1(lines),havingtp nkimpliest�3.Sowewanttondlinespassingthrough4points.4 Nextgureshowssetofalllinesgoingthroughatleast4points: Ifwewanttooutputallthese5lines,Qmusthavetotaldegreeatleast5.Buttheagreementparametert=4issmallerthan5,soQ(x;f(x))(foroneoftheselines;y=f(x))mightnotbe0.Notethatinthisexample,eachpointhastwolinescrossing.ThusanyQ(x;y)containingtheselinesasfactorsmustalsocontaineachpointtwice.NowwewilltrytounderstandwhatitmeansforQ(x;y)tocontaineachpointtwice.Example3IfQ(X;Y)2F[X;Y]hasrzeroesatpoint(0;0),thenithasnomonomialsoftotalweightlessthanr.SincewecantranslateQbyany( ;)2F2,wecangeneralizetheaboveexampletoaformaldenition:Denition10(Multiplicityofzeroes)ApolynomialQ(X;Y)issaidtohaveazeroofmulti-plicityw1atapoint( ;)2F2ifQ ;:=Q(x+ ;y+)hasnomonomialoftotaldegreelessthanw.5 ArmedwiththisdenitionandanideaonhowtoputadditionalconstraintsonQ,wewilllookatwhathappenstoLemmas7and8.Corollary11GivenpositiveintegersD;w1;:::;wnsuchthatkPi�wi+12�D+22,andpointsf( i;yi)gni=1F2,thefollowingholds:ThereexistsQ(X;Y)602F[X;Y]suchthatQ(x;y)hasazeroofmultiplicitywiat( i;yi)foralli,and(1;k)-weighteddegreeofQisD.MoreoversuchQcanbefoundbysolvingalinearsystem.Proof:NoticethatcoecientofxiyjinQ ;isXi0i;j0ji0ij0j i0�ij0�jqi0;j0:Hencewecanexpresswi-multiplicityconditionas�wi+12homogeneouslinearequations.Bypre-viouscalculation,weknowthatthenumberofQ'scoecientsis(D+1)(D+2) 2k.Thusanon-zerosolutionexistsifnPi�wi+12�D+22Lemma12Supposef( )=ifQ(x;y)hasr-zeroesat( ;).Then(x� )rQ(x;f(x)).Proof:WehaveR(x):=Q(x;f(x))=Q ;(x� ;f(x)�)=Q ;(x� ;f(x)�f( ))HereQ ;hasnomonomialsofdegreer.Also(x� )(f(x)�f( )).Foranynon-zeromonomialxiyjofQ ;,wehave(x� )i+j(x� )i(f(x)�f( ))j.Sincei+jr,wehave(x� )rQ ;(x� ;f(x)�)=Q(x;f(x)).Lemma13Ifadegreekpolynomialfhasf( i)=yifortvaluesofiandPi:f( i)=yiwi&#x-496;D,theny�f(x)Q(x;y)assumingQhas(1;k)weighteddegreeD.Proof:Iff( i)=yiandQ(x;y)hadamultiplicityofwizeroesat( i;yi)then(x� i)wiQ(x;f(x))bypreviouslemma.ThereforeQi:f( i)=yi(x� i)wiQ(x;f(x)).If(1;k)weighteddegreeofQ(x;y)isD,thenQ(x;f(x))0andy�f(x)Q(x;y).Puttingalltogether,weobtain:Theorem14Thealgorithmndsallpolynomialsf(x)suchthatPi:f( i)=yiwi&#x-496;q 2kPi�wi+12.Inparticular,ifwi=w,thenwecandecodefupton�terrorswitht�q nk�1+1 w.Corollary15(4)ForaReed-SolomoncodeofrateR,wecanlistdecodeupto1�p Rfractionoferrorsintimepolynomialintheblocklengthandtheeldsize.6 3SoftDecodingOnereasonwepickedeachwiindividuallyisbecausewe\encode"likelihoodinformation.Onemightthinkofwi'srepresentingacondencevalueonthereceivedsymbolandplacingmoreemphasisonsmallervalues.ThisconnectionhasbeenmadeformalbyKoetter-Vardy[5],whereitwasshownhowtochooseweightsoptimallybasedonchannelobservationsandchanneltransitionprobabilities.Oneimportantpointisthatthisapplicationrequiresdealingwithnon-negativerealweights.ThisismadepreciseinthefollowingTheoremwhichappearsin[2,Chapter6]:Theorem16Forallnon-negativerationalsfwi; gi2[n]; 2Fandnon-negativereal,thereisapoly�n;jFj;1 -timealgorithmtondalldegreekpolynomialsfsuchthatnXi=1wi;f( i)s kXi; w2i; +wmax:4ListDecodingofBinaryConcatenatedCodesThemainshortcomingoftheabovemethodsisthatthealphabetsizeispolynomialinblocklength.Wewilluseconcatenatedcodestoreducealphabetsize.AssumewearegivenaReed-SolomoncodeasouroutercodeCout:Fn2!Fn=R2withrateRandaninnercodeCin:Fm2!Fm=r2withrater=rate(Cin).Considerthebasiccomposition: ThiscodehasrateR0=R=r.Firstwewillstartwiththemostnaiveideaandworkourwayup.OurmaingoalinthissectionistogetabinarycodewhichisclosetoZyablov'sbound.4.1UniquelyDecodingInnerCodesConsideruniquelydecodingeachinnercodesandthenapplyinglistdecodingalgorithmontheoutercode.Therewillbeh�1(1�r) 2fractionoferrorsifaninnerblockisdecodedtoawrongcodeword.ThelistdecodingcapacityofCoutis1�q R r=1�p R0.Hencethisalgorithmcanlistdecodeupto�1�p R0h�1(1�r) 2-fractionoferrors.Howeverthisisquitebad,aswepickedupafactorof1 2thatweweretryingtoavoidbyusinglistdecodingattherstplace.7 Lemma17Givenpositivereals0r;R1,thereexistsabinarycodewhichcanbelist-decodedupto1�q R rh�1(1�r) 2fractionoferrorsinpolynomialtime.4.2ListDecodingInnerCodesInsteadofuniquelydecodinginnercodes,considerlistdecodingthem,by{say{bruteforce.RecallthatbyJohnsonBound,anybinarycodecanbelist-decodeduptoh�1(1�r�)-fractionoferrorswithalistsizeof`=`()O(1=).AslightcomplicationarisesonhowtoapplyReed-Solomonlist-decodingonamessagewhereeachsymbolisitselfanotherlist.Duringlist-decodingalgorithm,weonlyassumedall( i;yi)pairsweredistinct,andeverythingworkedneevenif i= jfordi erenti;jaslongasyi6=yj.Usingthisobservation,wecanapplyourlistdecodingalgorithmonthesetf( i;yij)gi;j.Thiswillreturnusalistofmessageswhichagreeswitht�p kNpointswhereNn`.Thereforethismethodwillgiveuslistdecodingupto1�q R rh�1(1�r�)fractionoferrors.Lemma18Givenpositivereals0r;R1andforanyreal0r,thereexistsabinarylinearcodewhichcanbelist-decodedupto1�q R rh�1(1�r�)fractionoferrors.Inordertogetabinarycodelistdecodableupto1� 2fractionoferrors,wecantake`=1= 2andr=O( 2),whichimplies=O( 2).BytakingR=r5,weobtain:Theorem19Foranyreal0 1,thereexistsalinearbinarycodeofrate ( 6)list-decodableupto1� 2fractionoferrorswithalistsizeofO1 3.Remark20Comparethistotheoptimalparameterofrate ( 2)andlistsizeO(1= 2).4.3WeightedListDecodingInnerCodesOneroomforimprovementtothepreviousalgorithmisthat,insteadoftreatingeachcodeinthelistequally,assigningdi erentweights.IfwerememberTheorem16,wecanobtainalistdecodingwitha(weighted)agreementofPni=1wi;f( i)q kPi; w2i; +wmax.Inordertominimizethis,weneedtoensurethattheweightsofinnercodesaroundapointshoulddecayin`2normrapidly.Althoughitisnotknownhowtogetthisforgeneralbinarycodes,Hadamardcodeshavethisproperty.5GoingbeyondReed-Solomoncodes&JohnsonradiusForReed-Solomoncodes,weshowedthatonecanecientlylistdecodeuptoaradius(fractionoferrors)equalto1�p R.Howeverweknowthatlistdecodinguptoafraction1�R�oferrors8 1 s::: n�s+1 1 y1ys+1yn�s+2 y2ys+2... 2 y3ys+3...... ... s�1 ysyn| {z }N=n=smanycolumnsIfwethinkofReed-Solomondecodingasinterpolatingapolynomialoveraplane(whichledtothep Rboundonagreementrequired),itmightseempossibletodecodewithagreementfractionaboutRs=(s+1)byinterpolatingin(s+1)-dimensions.Problem21WewanttondapolynomialQ(X;Y1;Y2;:::;Ys)602F[X;Y1;Y2;:::;Ys]suchthatQ( is;yis+1;yis+2;:::;y(i+1)s)=0for0in=s.6References1.S.Ar,R.Lipton,R.RubineldandM.Sudan,\Reconstructingalgebraicfunctionsfrommixeddata,"SIAMJournalonComputing,vol.28,no.2,pp.488{511,1999.2.V.Guruswami,\Listdecodingoferror-correctingcodes,"LectureNotesinComputerScience,no.3282,Springer,2004.3.V.GuruswamiandA.Rudra,\ExplicitCodesAchievingListDecodingCapacity:Error-CorrectionWithOptimalRedundancy,"IEEETransactionsonInformationTheory54(1):135-150(2008).4.V.GuruswamiandM.Sudan,\ImproveddecodingofReed-Solomonandalgebraic-geometriccodes,"IEEETransactionsonInformationTheory,vol.45,pp.1757{1767,1999.5.R.KoetterandA.Vardy,\Algebraicsoft-decisiondecodingofReed-Solomoncodes,"IEEETransactionsonInformationTheory,49(11):2809-2825(2003)6.M.Sudan,\DecodingReed-Solomoncodesbeyondtheerror-correctionbound,"JournalofComplexity,vol.13,no.1,pp.180{193,1997.10