Constacyclic Codes Over A Class Of Finite Commutative Rings

In this dissertation.the structure properties of constacyclic codes over a class of finite commutative rings are studied. In particular,the Gray-images of some constacyclic codes are discussed.Firstly,the ring R=R[u]/(ups-(pμ-1)>is proved to be a chain ring with maximumal ideal<1+u>.Similar to the discussions of S.X.Zhu and X.S.Kai in [67],we proved that(pμ-1)-constacyclic codes of length nps over the integer residue class ring R=Zpm correspond to the ideals of R[x]/(xn-u),where μ is a unit of Zpm,p a prime integer,n prime to p and m≥2a positive integer.gi(ζni) is proved to be accompanied by u+1,where ζ1i,ζ2i,…,ζhii in R[x]/<fi(x)>are the roots of fi(x)=0,fi(x),gi(x),i=1,2,…,k are monic irreducible polynomial over R,xn-u=f1,f2…fk,xn+1=g1g2…gk,fi(x)=gi(x),i=1,2,…,k. By using the relevant conclusions of ring isomorphsm,the structure of(pμ-1)-constacyclic codes over Zpm of arbitrary length over Zpm are discussed,two different classes of generator polynomials of such constacyclic codes of arbitrary Iength are obtaied and F=F0(x)+(1+u)F1(x)+…+(1+u)mps-1Fmps-1(x),where0≤ji≤mps,i=1,2…k1, l≤mps.We get the generator matrixes of the constacyclic codes of arbitrary length by using the generator polynomials of these constacyclic codes.If Fi(x)is accompanied by Fj*(x)when i+j=mps,i≥0.j≥0,then C=<F>is self-dual (pμ-1)-constacyclic codes over Zpm of arbitrary length.We also study(t+μλ)-constacyclic codes over a finite chain ring R of arbitrary length in a similar way, where μ is a unit of R,<λ>is the maximal idea of R,1≤t≤p-1and p is the characteristic of the residue field of R. The generator polynomials of the dual codes of(t+μλ)-constacyclic codes over a finite chain ring R of arbitrary length are obtained.Secondly, we study v-constacyclic codes over Fq+vFq,where v2=-1, q is a prime odd. The relations between v-constacyclic codes over Fq+vFq, ζ-constacyclic codes over Fq and-ζ-constacyclic codes over Fq are discussed, where ζ∈Fq,ζ2=-1.According to the relations,generator polynomials of v-constacyclic codes over Fq+vFq are discussed,the polynomial of a v-constacyclic codes over Fq+vFq of length n has the form of g(x)-2-1(1+ζv)91(x)+2-1(1-ζu)g2(x),where g1(x),g2(x)are monic polynomial over Fq,g1(x)|xn+ζ,g2(x)|xn-ζ.We define a class of Gray-maps on R[x]/(xn-v),Φs,t:R[x]/(xn-v)â†'Fq[x]/(x2n+1),Φs,t(a(x)+b(x)v)=ta(x)+sb(x)+xn(-sa+tb),where a(x),6(x)∈Fq[x],s,t∈Fq.If t2+s2≠0,it is proved that Φs,t is one to one mapping,and Φs,tÏ„=σΦs,t,where Ï„ is v-shift in Rn,σ is-1shift in Fq2n,so the Gray-images are linear nagacyclic codes of length2n over Fq.The Gray-image of the v-constacyclic code<g(x)>is<g1(x)g2(x)>.We prove that the Gray-maps are distance-invariant. The relation between the Gray-image of the dual code of a v-constacyclic code C of arbitrary length over Fq+vFq and the dual code of the Gray-image of the code C is Φ(Φ(C⊥))=(Φ(C))⊥,where φ:Râ†'R, Φ(a+bv)=a-bv.Thirdly,we study(1+2u)-constacyclic codes over R+uR and their Gray-images,where u2=-u.The relations between(1+2u)-constacyclic codes over R+uR,cyclic codes over R and negacyclic codes over R are discussed.According to the relations,we get the generator polynomial uG(x)+(1+u)F(x)of a(1+2u)-constacyclic code over R+uR,where F, G is the generator polynomials of the corresponding cyclic code over R and negacyclic code over R respectively. We define a class of Gray-maps on Rn:Φs,t(a(x)+6(x)v)=ta(x)+sb(x)+xn((t+2s)a(x)-sb(x)),where s,t∈R,a(x),b(x)∈R[x]. If s(t+s)is a unite in R, it is proved that Φs,t is one to one mapping,and the Gray-maps on(R+uR)" are distance-invariant and the Gray-images of(1+2u)-constacyclic codes over R+uR are linear cyclic code of length2n over R.We prove that the Gray-image of the dual code of a(1+2u)-constacyclic code C over R+uR is the dual code of the Gray-image of the code C,i.e.,Φs,t(C⊥)=Φs,t(C)⊥.Finally,we study v-constacyclic codes over R+vR of length n,where v2=-1,The relations between v-constacyclic codes over R+vR,ζ-constacyclic codes over R and-ζ-constacyclic codes over R are discussed,where ζ∈R and ζ2=-1. The generator polynomial of a v-constacyclic code over R+vR has the form of(1+ζv)G(x)+(1-ζv)F(x)>,where F=F1+λF2+...+λe-1Fe,G=G1+λe2+…+λe-1Ge are the generator polynomials of the corresponding ζ-constacyclic code and-ζ-constacyclic code respectively,where F0(x),F1(x)…Fe(x),Go(x),G1(x)…Ge(x)are monic pairwise coprime polyno-mials over R,F0(x)F1(x)…Fe(X)=xn-ζ,G0(x)G1(x)... e is the nilpotency index of A. We define a class of Gray-maps on (R+vR)n. It is proved that the Gray-maps are distance-invariant and the Gray-image of a v-constacyclic code (1+ζv)G(x)+(1-ζ(v)F(x)) of length n over R+vR is linear negacyclic codes of length2n over R, it’s generator polynomial is the relations between the Gray-image of the dual codes of v-constacyclic codes for length n over R+vR and the dual codes of the Gray-images are discussed.