Research On Perfact Codes Over R And Mass Formula Of Cyclic Codes Over S

Recently, in order to get—high quality linear codes, many reseachers have extended their research fields from finite fields to finite rings.The relationship between linear codes over finite fields and linear codes over finite rings has been set up through different Gray mappings.We can get a series of codes over finite fields with better parameters by using Gray mappings. In this paper, we discuss the existence of perfect linear codes over F_q+uF_q+u~2F_q+...+uk-1F_q with respect to homogenous metric and obtain the mass formula of cyclic code over the ring F_q+uF_q+vF_q+uvF_q.The details are given as follows:(1) In this paper, R denotes the ring Fg+uF_q+u~2F_q+...+uk-1F_q,where, uk= 0 and q is a power of a prime.The existence of perfect linear codes over F_q+uF_q+u~2F_q+...+uk-1F_q with respect to homogenous metric is studied, the sphere bounds of linear codes over R are obtained, By using the bounds we discuss the existence of perfect linear codes over R with respect to homogenous metric under two kinds of special circumstances.(2) We discuss the mass formula of cyclic codes of length n over the ring S= F_q+uF_q+vF_q+uvF_q,where u~2=0,v~2=0,uv= vu,(n,p)=1.By the Chinese remainder theorem, the problem translates to the classification of the ideals of the quotient ring (F_q+uF_q+vF_q+uvF_q)[x]/(f(x)),where f(x) is the basic irreducible polynomials over F_q+uF_q+vF_q+uvF_q.By classifying the ideals of the quotient ring R we obtain the mass formula of cyclic codes of length n over the ring S.