A Combining Method of Quasi-Cyclic LDPC

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1、IEEECOMMUNICATIONSLETTERS,VOL.9,NO.9,SEPTEMBER2005823ACombiningMethodofQuasi-CyclicLDPCCodesbytheChineseRemainderTheoremSehoMyungandKyeongcheolYang,Member,IEEEAbstract—InthispaperweproposeamethodofconstructingTheoutlineofthepaperisasfollows.InSectionII,wequasi-cycliclow-densityparity-check(QC-LDPC)c

2、odesoflargereviewQC-LDPCcodesanddefinenotationforourpresenta-lengthbycombiningQC-LDPCcodesofsmalllengthastheirtion.WeproposeamethodtoextendQC-LDPCcodesbasedcomponentcodes,viatheChineseRemainderTheorem.ThegirthontheCRTinSectionIIIandpresentafamilyofQC-LDPCoftheQC-LDPCcodesobtainedbytheproposedmethodis

3、alwayslargerthanorequaltothatofeachcomponentcode.codeswithno4-cyclesbyapplyingtheproposedmethodtoByapplyingthemethodtoarraycodes,wepresentafamilyofarraycodesinSectionIV.Finally,wegiveconcludingremarkshigh-rateregularQC-LDPCcodeswithno4-cycles.SimulationinSectionV.resultsshowthattheyhavealmostthesame

4、performanceasrandomregularLDPCcodes.II.QUASI-CYCLICLDPCCODESIndexTerms—ChineseRemainderTheorem,circulantpermu-tationmatrix,low-densityparity-check(LDPC)codes,quasi-AQC-LDPCcodeischaracterizedbytheparity-checkcyclic.matrixwhichconsistsofsmallsquareblockswhicharethezeromatrixorcirculantpermutationmatr

5、ices.LetCbetheI.INTRODUCTIONQC-LDPCcodesoflengthnL,whoseparity-checkmatrixisgivenbyOW-DENSITYparity-check(LDPC)codesfirstdiscov-⎡⎤LeredbyGallager[3]havearemarkableperformancewithPa11Pa12···Pa1(n−1)Pa1niterativedecodingthatisveryclosetotheShannonlimitover⎢Pa21Pa22···Pa2(n−1)Pa2n⎥⎢⎥additivewhiteGaussia

6、nnoise(AWGN)channels[5],[6].MostH=⎢........⎥(1)⎣..···..⎦methodsfordesigninggoodLDPCcodesarebasedonrandomPam1Pam2···Pam(n−1)Pamnconstruction,butalgebraicallystructuredLDPCcodesmaybeneededforimplementationpurposes.InthecaseofrandomwhereP=(Pij)istheL×LpermutationmatrixdefinedbyLDPCcodesoflargecodelength

7、,asignificantamountofmemoryisneededtostoretheirparity-checkmatrices.Also,1ifi+1≡jmodLPij=(2)itishardtoaccessthememoryandencodedataefficiently.0otherwiseQuasi-cyclicLDPC(QC-LDPC)codesmaybeagoodandaij∈{0