使用MATLAB的有限域计算

1、GaloisFieldComputationinMATLAB:PrimitivePolynomial:AnIrreduciblepolynomialp(X)ofdegreemissaidtobeprimitiveifthesmallerpositiveintegernforwhichp(X)dividesXn+1isn=2m一1.Forexample,p(X)=X4+X+1dividesX15+1butnotdividesanyXn+1for1n15.InMatlabyoucaneasilyfindtheprimitivepolynomialsforanydegreeusingprimpoly

2、(m)function.Example:m=4;Definem=4forGF(24)s=primpoly(m)Outputis:Primitivepolynomial(s)=dt+dt+is=19Hereitshowstheprimitivepolynomialandanintegerwhosebinaryrepresentationindicatesthecoefficientsofthepolynomial.Notethattherecouldbemorethanoneprimitivepolynomialforaparticulardegreeofm.GaloisFieldarithme

3、tic:TodemonstrateGaloisFieldarithmeticweconsiderfollowingtablefordegreem=4anprimitivepolynomialp(X)=1+X+X4PowerRepresentationPolynomialRepresent4-TupleRepresentation000000111000aa0100a2a20010a3a30001a41+a1100a5a+a20110a6a2+a30011a71+a+a31101a81+a21010a9a+a30101a101+a+a2111013aiia+a2+a30111ai21+a+a2+

4、a31111ai31+a2+a31011ai41+a31001TheMatlabfunctiongftuple(),SimplifyorconverttheformatofelementsofaGaloisfield.Thatmeans,youcanfindthetuplerepresentationofcorrespondingpowerrepresentationbygftuple()function.Example:X=gftuple(11,4);Y=gftuple(14,4);Theoutputis:X=0111Y=1001InMatlabyoucandoanyGaloisFielda

5、rithmeticusinggf()function.Example:X=gf(6,4)%firstargumentisintegerequivalentoftuple%representationand2ndargumentisdegreemY=gf(13,4)Z=X+YTheoutputis:X=GF(2人4)array.Primitivepolynomial=DA4+D+1(19decimal)Arrayelements=6Y=GF(2A4)array.Primitivepolynomial=DA4+D+1(19decimal)Arrayelements=Z=GF(2人4)array.P

6、rimitivepolynomial=DA4+D+1(19decimal)Arrayelements=11Note:Thearrayelementsareshownintuplerepresentationformat.Tounderstandityoumayneedtoconvertfromtuplerepresentationtopowerrepresentationfromthetable.Intheaboveexamplewedidtheadditionofa5+a7whichisa13.Similarlyyoucadomultiplication,divisionorsubtract

7、ion.MinimalpolynomialsMinimalpolynomialsofelementsinGF(24)generatedbyp(X)=X4+X+1aregiveninfollowingtable.ConjugateRoots0X1X+1a,a2,a4,a8a3,a6,a9,a12a5,a10a7,a11,a13,a14Minimalpolynomials(X)X4+X+1X4+X3+X+1X2+X+1X4+X3+1Byusingminpol()functioninmatlabyoucangenetaretheminimalpolynomialforanyroot.Againusi

8、ngroots()functionyoucanfindtheconjugaterootsforaparticularminimalpolynomial.Example:m=4;e=gf(6,4);em=minpol(e)%Findminimalpolynomialofe.emisin%GF(2Am)emr=roots(gf(00111,m)%RootsofDA4+DA3+1inGF(2Am)Theoutputis:em=GF(2)array.Arrayelements=emrGF(2人4)array.PrimitivepolynomialDA4+D+1(19decimal)Arrayeleme

9、nts=9111314GenerationofgeneratorpolynomialofBCHcodes:ForBCHcodesgeneratorpolynomialscanbecalculatedbyfollowingequationasg(X)=LCMp(X),O(X),O(X),O(X)升,X)whereO(X)arethe13572t-1iminimalpolynomials.Nowform=4,wecancalculatethegeneratorpolynomialfor(15,11),(15,7)and(15,5)BCHcodesasfollowing:g1(X)=O1(X)for(15,11)BCHcode,t=1g12(X)=LC1MO1(X),O3(X)for(15,7)BCHcode,t=2g23(X)=LCMO11(X),O33(X),O5(X)for(15,5)BCHcode,t=3Inmatlabyoucaneasilygeneratethegeneratorpolynomialbyusingbchgenpoly()function.Example:genpoly1=bchgenpoly(15,11)genpoly2=bchgenpoly(15,7)Theoutputis:genpoly=GF(2)array.Arrayelements=10011gen