Research On Codes Over Finite Rings

In this paper,based on the previous research theory of coding,I dis-cussed and studied the generation structure and excellent properties of self-dual code,LCD double cyclic code,DNA code and linear code on a specific ring,and committed to finding codes with good parameters.The research content specifically includes the following three aspects.1.Double circulant codes of length 2n over the semi-local ring Fq+uFq+vFq+uvFq,u2=v2=0.uv=vu are studied when q is a prime power.Exact enumerations of self-dual and LCD double circulant codes for a positive integer n are given.Using a distance-preserving Gray map,self-dual and LCD codes of length 8n over are constructed when q is even.Using random coding and the Artin conjecture,the modified varshamov-Gilbert bounds were derived on the relative distance of the codes considered.2.In this section,we construct cyclic DNA codes over the ring R2=F2[u,v]/{u3,v2-v,vu-uv).The correspondence between the elements of R2 and the alphabet {A,T,G,C}3 is obtained by a given Gray map.Moreover,some properties of binary images of the DNA code under the Gray map are also discussed.Finally,two examples of cyclic DNA codes over R2 are presented to illustrate the obtained results.3.Linear codes over finite rings are described here as trace codes.Cyclic codes over Galois rings are given a trace description as well.The main tools are the notion of trace dual bases,in the case of linear codes,and of normal bases of an extension ring over a ring,in the case of cyclic codes.