Research On The Depth Distribution Of Linear Codes Over Finite Chain Rings

Along with the further research on the structure and properties of linear codes over rings, researchers have generalized the depth of linear codes over finite fields to the depth of linear codes over rings. In this paper, we mainly study the depth spectrum and distribution of linear codes over rings from the point of generator polynomials and generator matrices of linear codes. The details are given as follows:Firstly, we give the concept of depth of codewords over R=Fpk+uFpk. By using of the structure of cyclic codes over this ring and the generator polynomials of cyclic codes over finite fields, the structure and properties of linear codes r0 (C) and r1 (C) are determined, then depth spectrum and depth distribution of cyclic codes of an arbitrary length over ring R are obtained. Finally, the existence conditions for cyclic self-dual codes are also discussed.Secondly, based on the study of depth distribution of cyclic codes, we study the depth spectrum of negacyclic codes and their self-dual codes of an arbitrary length over ring R. The depth distribution of negacyclic codes of length pm is given.Thirdly, we give the definition of linear codes r0(C), r1(C) and r2(C) over ring R= Fq+uFq+u2Fq. According to the computing methods for depth spectrum of linear codes over finite fields, the depth spectrum and distribution of negacyclic codes over R are determined, the computing formulas for depth distribution of self-dual negacyclic codes are also given.Finally, the depth spectrum of linear code Cover ring R’=F2+uF2+u2F2+u3F2 is studied. The generator matrices of r0(C),r1(C),r2(C)andr3(C)are obtained, making use of the standard generator matrix of linear code Cover R’. It is proved that there are at least k0+k1+k2+k3 different nonzero values in the depth spectrum of C.