RS码编译码算法的实现(学习RS码极为有用)

:049026756(2000)0620868205RS,,(,610064):RS,RS,.:RS;;;;:TN914.31:ARS,,,,RS,RS.,RS,,,.1RS,p,qp,GF(q)q.q(n,k)xn-1,GF(q)n-k.sv,n=qs-1qBCH.2svv.s=1qBCHRS.BCH.RS,.RS,n-k+1.RS,,,.RS,k.(n,k)RS,km,k,m.vRS:n=2m-1kn-k=2vd=2v+1m(2m-1)kmm(n-k)m(2v+1)vRSg(x)=(x+a)(x+a2)(x+a3)(x+a4)(x+a2v)(1):2000210223200012376()JournalofSichuanUniversity(NaturalScienceEdition)Dec.2000Vol.37No.6aGF(2m).,g(x)a,a2,,a2v.GF(2m).g(x)(n,n-2v).g(x),GF(2m)n-1.RS01,RS,RS.2RSRS,,.mk-1xk-1+mk-2xk-2++m0xn-kg(x),r(x),xn-k,RS.RS,vg(x)=2vi=1(x+ai)=2vi=0gixi(2)aGF(2m).m(x),c(x),c(x)=xn-km(x)+xn-km(x)modg(x)(3)3RS,.,RS,RS,2.RS,,.RS.3.1SkT(x)=t0+t1x++tn-1xn-1.R(x)=r0+r1x++rn-1xn-1.e(x)=R(x)-T(x)=e0+e1x++en-1xn-1ei=ri-viGF(2m).e(x)v()x1,x2,,xv,e(x)=e1x1+e2x2++evxve(x),xiei.2vaiR(x),sk=0,;sk0,sk.R(ai)=e(ai)+T(ai)=e(ai)(T(ai)=0),S1=R(a)=e1x1++evxvS2=R(a2)=e1(x1)2++ev(xv)2S2v=R(a2v)=e1(x1)2v++ev(xv)2v(4)9686:RS3.2eixi,,,.vxi,vxiv(xi)=xiv+1xiv-1++v=0i=1,2,,v(5),xi,eixji,eixiv+j+1eixiv+j-1++veixji=0i=1,2,,v(6)(6)i=1,2,,v,(4)Sv+j+1Sv+j-1++vSj=0(7)(7),vRS,2vS1,S2,,S2v.(7),j1v,v.Sv+1+1Sv++vS1=0Sv+2+1Sv+1++vS2=0S2v+1S2v-1++vSv=0(8),(x)1v.,.,.3.3v,,,,,(xi)=xti+1xt-1i++t=0i=1,2,,t,an-1,ai(i=n-1,n-2,,0)(ai),i..3.4(4),1v.3.5,,.4RS.,7,7.1BCH(15,11).g(x)=m1(x)=x4+x+1078()37H=111101011001000011110101100100001111010110010111010110010001H=[a14a13a12a2a1a0],ai,,.,(2).a14+a12=[1001]T+[1111]T=[0110]T=a5,n=2m-1.a5a11=a16=a1mod152,15RS.n=15,t=2m=4,r=11,d=5.RS(15,11),g(x)=(x+a)(x+a2)(x+a3)(x+a4)=x4+a13x3+a6x2+a3x+a104.1m=(a2000a30000a90),m(x)=a2x10+a3x6+a9x,T(x)=xn-km(x)+xn-km(x)modg(x)=a2x14+a3x10+a9x5+x3+a3x+a13:T=(a2000a30000a90a00a3a13).4.2R2,511,a9a6,0a,:R=(a200aa30000a60a00a3a13)R(x)=a2x14+ax11+a3x10+a6x5+x3+a3x+a13(1)S1=R(a)=a2a14+aa11+a3a10+a6a5+a3+a3a+a13=a16mod15+a12+a13+a11+a3+a4+a13=[0010]T+[1111]T+[1101]T+[1110]T+[1000]T+[0011]T+[1101]T=a3S2=R(a2)=a2,S3=R(a3)=a8,S4=R(a4)=a5,S5=R(a5)=a12,S6=R(a6)=a13(2)7RS,.Sk=0,.D3=S1S3S5+S1S24+S22S5+S330,3;D3=0D2=S1S3+S220,;D3=D2=0S10,.,D3=S1S3S5+S1S24+S22S5+S33=0D2=S1S3+S22=a11+a4=a13..(8)1=(S1S4+S3S2)/D2=(a3a5+a8a2)/a13=(a8+a10)/a13=([0101]T+[0111]T)/a13=a1/a13=a32=(S2S4+S23)/D2=a1786:RS(3):(x)=x2+a3x+a.a14,a13,,a0,x=a5a11(x)=0,a5a11511.(4)x1=a5,x2=a11(4),e1=(s1x2+s2)/(x1x2+x21)=(a3a11+a2)/(a5a11+a10)=(a11+a2)/(a5a11+a10)=a13/a8=a5e2=(s1x1+s2)/(x1x2+x22)=a(5)5a511a,a5a511,a5+a6=[0110]T+[1100]T=[1010]T=a9a+a=[0010]T+[0010]T=0(a2000a30000a90a00a3a13),,.:[1],.SDH[M].:,2000.[2],.[M].:,1996.[3].[M].:,1988.THEIMPLEMENTATIONOFRSENCODINGANDDECODINGTAODe2yuan,HEXiao2hai,WUZhi2hua(CollegeofElectronicInformation,SichuanUniversity,Chengdu610064,China)Abstract:Ofallkindsoferror2controlencodingtechniquesincommunicationsystem,RScodeiswidelyusedforitshighabilityinerrorcorrection.OnthebasisofanalysingtheprincipleandqualityofRScode,deducesthecourseofitsencodinganddecodingandexplainedthedetailedimplementationmethod.Keywords:RScode;GFregion;nonbinarycode;encoding;decoding278()37