Research On Some Problems Of Codes Over The Ring F_q+uF_q+...+u^k-1F_q

Many properties of codes have been studied over the Galois extension of finite chain ring Fp+uFp (P is prime),we’ve got the important conclusion that trace codes of dual codes are the dual of subring subcodes over this Galois extension, this theorem corresponds with Delsarte’s famous subfield theorem,at the same time all ring automorphisms on this Galois extension are obtained. Trace representation of irreducible cyclic codes of special length on residue class ring Zq (where q=pa, P for prime, a is any positive integer) has been given in relative articles, this has an important role to determine the weight distribution ofirreducible cyclic code over the ring.The relative properties of codes over the finite chain ring R= Fq+uFq+...+uk-1Fq,where q=pa,P is prime, a∈N and uk=0,has been widely researched.But,until now,no one has researched the Galois extension of this ring and also nothing has done about the trace representation of odd length irreducible cyclic codes over this ring.This thesis has just worked on this two problems.First of all,some theories about the Galois extension of the ring R=Fq+uFq+...+uk-1 Fq are given where q= pa,p is prime,a∈N,and uk=0· Relative properties of this Galois extension has been researched.Trace codes and subring subcodes over the Galois extension are defined,and it is proved that the trace codes of dual codes are the dual of subring subcodes,also all ring automorphisms on this Galois extension are obtained.Moreover,a kind of linear codes,which the relative parameters are[pma-1,m,(pa-1)pa(m-1)] over the ring R are given.Also,a generating polynomial of odd length cyclic codes over ring R= Fq+uFq+...+uk-1Fq is given.it is proved that the ring R[x] / (xn-1) is principal ideal ring.Moreover,trace representation of irreducible cyclic codes of length n= qm-1 over the ring R is given.