Research On Linear Codes Over Several Finite Non-chain Rings

With the development of coding theory, coding theory research over finite rings has an important significance from theory to application. Based on the previous excellent working on coding theory over finite rings, this thesis makes a deepgoing research on some classes of linear codes including double cyclic codes over finite chain ring, Mac Williams identities of linear codes, depth spec-trum and skew cyclic codes over finite non-chain rings. The details are given as follows:1、The Mac Williams identities with respect to RT metric for linear codes over Mn×s(Rk) are studied. The Lee complete p weight enumerator and the ex-act complete p weight enumerator over the ring are defined, and the Mac Williams identities with respect to RT metric for these two weight enumerators of linear codes over the ring are obtained.2、We study the algebraic structure of the double cyclic codes over Fq+ uFq+u2Fq. We determine the generator polynomials of the double cyclic codes and give the minimal generating sets of this family of codes. Finally, we show the relationship of generators between the double cyclic code and its dual.3、We investigate the depth spectrum and the depth distribution of linear codes over F2+uF2+u2F2. By using homomorphism of abelian groups from-R2= F2+uF2+u2F2 to F2 and the generator matrices of linear codes over R2, the upper bound and lower bound of the depth spectra of linear codes of type 8k14k2k3 are obtained. We also give the depth distribution of a linear code over R24、Skew cyclic codes over the ring Fq+uFq+vF1+uvF1 are studied. According to the Chinese remainder theorem, we discuss the structural prop-erties of skew cyclic codes over this ring and give a formula for the number of skew cyclic codes of length n over the ring. We study the skew cyclic codes over Fq+vFq+…+vm-1Fq. We give the generator polynomials of skew cyclic codes over this ring and investigate the structural properties of skew cyclic codes over the ring by a decomposition theorem. The generator polynomial of the dual of a skew cyclic code is obtained. The idempotent generators of skew cyclic codes over Fq and Fq+vFq+…+vm-1Fq are considered. Moreover, we specially discuss the structural properties of Fq+vF1+v2F1, that is m = 3.