MacWilliams Identities of Linear Codes over Ring Mnxs (R) with Respect to the Rosenbloom-Tsfasman Metric

The Rosenbloom-Tsfasman metric (RT, or ρ, in short) is a non-Hamming metric and is a generalization of the usual Hamming metric, so the study of it is very significant from both a theoretical and a practical viewpoint. In this study, the definition of the exact complete ρ weight enumerator over Mnxs (R) is given, where, R = Fq+uFq+...+ut-1Fq and ut = 0 and a MacWilliams type identity with respect to this RT metric for the weight enumerator of linear codes over Mnxs (R) is proven which generalized previous results. At the end, using the identity, the MacWilliams identity with respect to the Hamming metric for the complete weight enumerator cweC (x0, x1, xu,..., x(q-1)+(q-1)u+...+(q-1)ut-1) of linear codes over finite chain ring R is derived too.