Research On The Linear Codes And Their MacWilliams Identity Over Two Classes Of The Finite Non-chain Rings

In this paper, we mainly study the the linear codes and their MacWilliams identity over two classes of the finite non-chain ring. The details are given as follows:(1) We study the linear codes and their MacWilliams identity over the ring R= Z4+vZ4(v2=v). First we give the propositions of Gray maps and several projections of linear codes over ring R,and consider the relationship of minimum Lee weight between the linear codes over ring R and the corresponding codes which are obtained by means of the projections. Next, by defining the Gray weight enumerator and symmetrized weight enumerator, we could establish the MacWilliams identities between linear codes and their dual over ring R with respect to the Gray weight enumerator, the symmetrized weight enumerator and Lee weight enumerator.(2) We try our best to study the linear codes and their MacWilliams identity over the ring Rk,m=Fq[u,v]/<uk,vm,uv-vu>,which q is a power of the prime p and k and m are both greater than 1.First we identity Lee weight and introduce a distance and duality preserving Gray map from Rk,m to Fkmq together with the Lee weight.Next we prove the MacWilliams identities for codes over Rk,m for all the relevant weight enumerators,especially the MacWilliams identities between linear codes and their dual over ring Rk,m with respect to Lee weight enumerator is obtained.