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1、IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.55,NO.7,JULY20093051ChannelPolarization:AMethodforConstructingCapacity-AchievingCodesforSymmetricBinary-InputMemorylessChannelsErdalArıkan,SeniorMember,IEEEAbstract—Amethodisproposed,calledchannelpolarization,A.Prelimin
2、ariestoconstructcodesequencesthatachievethesymmetriccapacityWewritetodenoteagenericB-DMCwith ofanygivenbinary-inputdiscretememorylesschannel(B-DMC).Thesymmetriccapacityisthehighestrateachiev-inputalphabet,outputalphabet,andtransitionprobabilitiesablesubje
3、cttousingtheinputlettersofthechannelwithequal.Theinputalphabetwillalwaysbeprobability.Channelpolarizationreferstothefactthatitispos-,theoutputalphabetandthetransitionprobabilitiesmaysibletosynthesize,outofindependentcopiesofagivenB-DMCbearbitrary.Wewritetode
4、notethechannelcorresponding ,asecondsetofbinary-inputchannels tousesof;thus,withsuchthat,asbecomeslarge,thefractionofindicesforwhich . isnearapproaches andthefractionforwhich isnearapproaches0 .ThepolarizedchannelsGivenaB-D
5、MC,therearetwochannelparametersofpri- arewell-conditionedforchannelcoding:oneneedonlymaryinterestinthispaper:thesymmetriccapacitysenddataatratethroughthosewithcapacitynearandatratethroughtheremaining.Codesconstructedonthebasisofthisideaarecalledpola
6、rcodes.Thepaperprovesthat,givenanyB-DMCwith andanytargetrate ,thereexistsasequenceofpolarcodes suchthathasblock-lengthandtheBhattacharyyaparameter,rate,andprobabilityofblockerrorundersuc-0cessivecancellationdecodingboundedas in
7、dependentlyofthecoderate.Thisperformanceisachievablebyencodersanddecoderswithcomplexity foreach.Theseparametersareusedasmeasuresofrateandreliability,IndexTerms—Capacity-achievingcodes,channelcapacity,channelpolarization,Plotkinconstruction,polarcodes,Reedr
8、espectively.isthehighestrateatwhichreliablecommu-Muller(RM)codes,successivecancellationdecoding.nicationispossibleacrossusingtheinputsofwithequalfrequency.isanupperboundontheprobabilityof
