The MacWilliams Identity Over A Class Of Finite Rings

Codes over rings have recently received a great deal of interest among coding theorists. Because some excellent binary nonlinear codes can be obtained from the binary images of the linear codes over rings under a nonlinear map. In this paper, the Mac Williams identity betweenthe linear code and its dual over the ring F_p + uF_p+…+u^k-1F_p is discussed. A symmetrizedweight enumerator of the linear code over the ring F_p + uF_pis defined in this paper. By using the discrete Hadamard transformation, a symmetrized Mac Williams identity between the linear code and its dual over the ring F_p + uF_p is given.Furthermore, the above results is generalized to the ring F_p +…+ u^k-1F_p. The symmetrized weight enumerator of the linear code over thering F_p + uF_p+…+u^k-1F_p is defined, and the symmetrized Mac Williams identiy between thelinear code and its dual over the ring F_p + uF_p +…+u^k-1F_p is also given by using a similardiscrete Hadamard transformation.