Research On The Algebraic Structure Of Double Cyclic Codes Over A Class Of Non-chain Rings

As a generalization of the finite field case,coding over finite rings has become a hot research topic in recent years because of its good algebraic structure and properties.This paper studies the double cyclic codes over a special class of finite rings.We get the algebraic structure of such class of double cyclic codes over finite non-chain rings.The specific research content can be divided into the following three parts:1.This dissertation first analyzes the algebraic structure of polynomial ring over F_q+vF_q,v2=v.Using the factorization over this ring,the maximal common factor and the divisibility theory of polynomials,the structure of the single cyclic codes over F_q+vF_q,v2=v is explained and illustrated from a new perspective,including the generating polynomials,the generators of dual codes and relationship between them.Simultaneously,the duality of single cyclic codes over this ring is fully explained.2.Based on the good algebraic structure of F_q+vF_q,v2=v and the theory of polynomials over single cyclic codes,we obtain some new results for double cyclic codes over F_q+vF_q,v2=v.This includes generating polynomials,minimal generating sets,generating matrices of double cyclic codes and the dual codes generation polynomials.The relationship between dual codes generators and the proposed codes generators is also explained clearly.3.Considering the case of the double cyclic codes over F_q+vF_q+v2F_q(v3=v),we not only obtain the conclusion similar to that of the double cyclic codes over F_q+vF_q(v2=v)but also obtain the core technique' of this paper.We construct a set of idempotent orthogonal bases from the generating relations of the generators in the ring.Then,we use the excellent divisibility of this basis in polynomial theory to obtain various algebraic properties of the double cyulic codes over F_q+vF_q+v2F_q(v3=v).Based on this treatment,we make further generalizations to get the algebraic structure of double cyclic codes over a class of finite non-chain rings.It is the F_q+vF_q+…+vs-1F_q-double cyclic codes,where vs=v,s?2.It is obvious that F_q+vF_q-double cyclic codes and F_q+vF_q+v2F_q-double cyclic codes are both exceptional cases of such codes.