CYCLIC CODES OVER RINGS OF MATRICES

In this paper, we consider the ring of matrices A of order 2 over the ring F-2[u]/(u(k)), where u is an indeterminate with u(k) = 0, i.e. A = M-2(F-2[u]/(u(k))). We derive the structure theorem for cyclic codes of odd length n over the ring A with the help of isometry map from A to F-4[u, v]/(u(k), v(2), uv- vu), where v is an indeterminate satisfying v(2) = 0 and uv = vu. We define a map theta which takes the linear codes of odd length n over A to linear codes of even length 2kn over F-4. We also define a weight on the ring A which is an extension of the weight defined over the ring M-2(F-2). An example is also given as applications to construct the linear codes of odd length n over A.