LDPC码

MYUNGandYANG:ACOMBININGMETHODOFQUASI-CYCLICLDPCCODESBYTHECHINESEREMAINDERTHEOREM

825

Fig.1.PerformanceofsomeregularQC-LDPCcodesconstructedwiththeexponentcombiningbytheCRT.

girthlargerthanorequaltothoseofC1andC2.Theorem2canbedirectlyextendedtothegeneralcasewithmorethantwoQC-LDPCcodes.

IV.AFAMILYOFQC-LDPCCODESWITHNO4-CYCLESThecombiningbytheCRTcanbeusedtoconstructalargeQC-LDPCcodewithno4-cyclesfromsmallQC-LDPCcodesascomponentcodes.Here,onlyarraycodeswillbeconsideredascomponentcodes,eventhoughotherQC-LDPCcodescanbenaturallyemployed.

TheexponentmatrixEofanarraycodeC(p,m)isde nedbyEij=(i 1)(j 1)for1≤i≤mand1≤j≤p[1].Herepisprimeandmisapositiveintegersuchthatm≤p.Theparity-checkmatrixH(p,m)ofC(p,m)canbeobtainedbyexponentcouplingofEandthep×pcirculantpermutationmatrixPin(2).Itiswell-knownthatthegirthofC(p,m)is6[1],[9].

Theorem3:LetC(p,m)bethearraycodewithparity-checkmatrixH(p,m)foraprimepandapositiveintegerm≤p.Fori=1,2,letpibeaprime(p1=p2)andE(Hi)bethem×p1p2exponentmatrixofHi,respectively,where

H1=[H

(p1,m)H(p1,m)···H(p1,m)]p ,

2times

H2=[H

(p2,m)H(p2,pm)···H(p2,m)].times

1LetEbethem×p1p2matrixoverZE(Hp1p2obtainedby

combiningE(Hmp1)and2)basedontheCRTandletbethe2

H

ofEandthe1p2×p2p1p2matrixobtainedbyexponentcoupling1p2×p1p2circulantpermutationmatrixin(2).ThentheQC-LDPCcodewithparity-checkmatrixHhasno4-cycles.

Proof.Letr1andr2betheleastpositiveintegerssatisfying(6)and(7)for4-block-cyclesinHp1andH2,respectively.Sincethe(i+j11)thandthe(i+j2p1)thcolumnblocksin

H1arethesameforjareno1=j4-cycles2,thereexist4-cyclesbetweenthem.But,therebetweenthe(i1+j1p1)thandthe(i2+jotherhand,2pthe1)thcolumnblocksinH(i+j1fori1=icolumn2.OntheblocksinH1p1)thandthe(i+j2p1)th(jj2aredistinctsince(i+j.Therefore,there1p1) (i+jareno4-cycles2p1)=between1 2)pthe1=0modp(i+j21p1)thandthe(i+jandH2p1)thcolumnblocksinH2.Thisimpliesthatr1r2>1hasno4-cyclesbyTheorem2. ThemethodgiveninTheorem3cannotbedirectlyusedtoconstructlow-rateQC-LDPCcodesoflargelength,sincethenumberofrowblocksareconstant.However,itmaybepossibletoconstructlow-rateQC-LDPCcodeswithoutshortcyclesinasimilarway.NotethatthecombiningmethodbasedontheCRTcanbeappliedtoconstructingQC-LDPCcodes,regardlessofwhethertheyareregularornot.

Figure1showsthebiterrorrate(BER)andframeerrorrate(FER)performanceoftheshortenedQC-LDPCcodescon-structedbythecombiningmethodviatheCRToveranAWGNchannel.Here,(N,K,j)(p1,p2)denotes(codelength,infor-mationlength,columnweight)(arraycodesC(p1,j),C(p2,j)).Theyhavelittleperformancedegradationduetotheirrestrictedstructure,ascomparedwithrandomlyconstructedregularLDPCcodeswiththesameparameters.Inthecasethatthecolumnweightislargerthan3,therearenoseriouserror oorsatFER=10 4.

V.CONCLUDINGREMARKS

WediscussedhowtocombineQC-LDPCcodesofsmalllength,basedontheCRT.WeshowedthatthemethodcanbeusedforconstructingQC-LDPCcodeswithno4-cycles.Byapplyingthemethodtoarraycodesascomponentcodes,wepresentedafamilyofQC-LDPCcodeswithno4-cycles.ThisapproachcanbedirectlyextendedtootherQC-LDPCcodes.

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