Large Girth Quasi-Cyclic LDPC Codes Based on the

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1、342IEEECOMMUNICATIONSLETTERS,VOL.13,NO.5,MAY2009LargeGirthQuasi-CyclicLDPCCodesBasedontheChineseRemainderTheoremXueqinJiangandMoonHoLee,SeniorMember,IEEEAbstract—Inthisletter,weconsidertwoproblemsassociatedThisletterisorganizedasfollows.SectionIIreviewQC-withquasi-cycliclow-densityparity-che

2、ck(QC-LDPC)codes.LDPCcodes,arraycodesandshortenedarraycodes.TheThefirstishowtoextendthecodelengthofaQC-LDPCcodegeneralizedcombiningmethodviaCRTisgiveninSectionIII.withoutreducingthegirth.ThesecondishowtodesignaQC-WeproposeamethodtoextendashortenedarraycodeintoLDPCcodewithaprescribedgirtheasil

3、y.WedealwiththesetwoproblemsbyusingacombiningmethodofQC-LDPCcodesdifferentlengthsbasedontheCRTinSectionIV.AmethodviatheChineseRemainderTheorem(CRT).CodesconstructedforthedesignofaQC-LDPCcodewithaprescribedgirthiswithourproposedmethodhaveflexiblecodelengths,flexiblegiveninSectionV.Finally,Secti

4、onVIconcludestheletter.coderatesandlargegirth.Simulationresultsshowthattheyperformverywellwiththeiterativedecoding.II.QUASI-CYCLICLDPCCODESIndexTerms—LDPCcodes,quasi-cyclic,arraycodes,short-enedarraycodes,ChineseRemainderTheorem(CRT),girth.AQC-LDPCcodeischaracterizedbyitsparity-checkmatrix⎛⎞

5、Pa00Pa01···Pa0(n−1)I.INTRODUCTION⎜Pa10Pa11···Pa1(n−1)⎟OW-densityparity-check(LDPC)codes,firstproposedH=⎜⎝⎟⎠,(1)············Lintheearly1960’s[1]andre-discoveredin1996[2],Pa(m−1)0Pa(m−1)1···Pa(m−1)(n−1)haverecentlyattractedmuchattentionduetotheircapacity-approachingperformanceandlowdecodingcomp

6、lexity.Onewhereaij∈{0,1,...,L−1,∞}andPaijfor0≤i≤parameterthatisusuallytargetedforoptimizationinthem−1,0≤j≤n−1representsanL×LzeromatrixprocessofdesigningLDPCcodesisthegirthoftheunderlyingifaij=∞andrepresentsanL×LcirculantpermutationTannergraph[1][3][4].Foriterativesum-productdecodingofmatrixo

7、btainedbycyclicallyshiftingtherowsoftheidentityquasi-cyclic(QC)LDPCcodes,apartofperformancelossmatrixtotherightbyaijtimesotherwise.Themblock-rowsisattributedtosmallvaluesofgirth.ArraycodeswhichcanofHcanbeindexedbytheintegersfrom0tom−1,andbeviewedas