Codes Over A Class Of Rings And Their Depth Distributions

With the advancement of communication technology in modern society,the coding theory is developing rapidly.In 1994,Studies by Hammons et al.showed that some nonlinear codes with excellent performance could be considered as Gray images of some codes over finite rings.Their findings expand the research fields of linear codes,so people expect to make more certain good codes over finite rings.In the recent ten years,with the continuous improvement of the coding theory over finite chain rings,people begin to pay close attention to studying codes over finite non-chain rings.Due to the limitations of finite non-chain rings' structures,it is difficult to study codes over them,so far,there is not a lot of research results on them.Self-dual codes is a special class of linear codes,and there are many methods to construct them.Because of their good properties,self-dual codes have always been a research hotspot.On the other hand,the depth of codewords is an important indicator which can represent the complexity of codewords,and it is also an important tool to study the linear complexity of sequences.It plays an important role for the construction and classification of the linear codes.Therefore,studies of self-dual codes and codewords' depth are of great theoretical significance.In this paper,we will study codes over the finite non-chain ring R and the depth distribution of these code(where R= Fq + vFq+v2Fq(v3 = v)).Firstly,By constructing Gray map and related projections,we can establish relationships between codes over R and Fq,and give a method of constructing linear codes and their self-dual codes over R.We also determine the relationship between the number of self-dual codes over Fq and R,and give the necessary and sufficient conditions for the existence of self-dual codes over R.Secondly,we study some properties of the depth of codewords over R and obtain the recursive algorithm of computing the depth of codewords.We prove that the depth spectrum and depth distribution of linear codes over R can be induced into the depth spectrum and depth distribution of linear codes over Fq.Then,some properties of codes over Fq on the depth distribution are discussed.