Concatenated structure of cyclic codes over of length 4n

Let where n is odd and k a positive integer. We present a canonical form decomposition for every cyclic code over of length N, where each subcode is concatenated by a basic irreducible cyclic code over of length n as the inner code and a constacyclic code over a Galois extension ring of for length as the outer code. For the case of , by determining their outer codes, we give a precise description for cyclic codes over , present dual codes and investigate self-duality for cyclic codes over of length 4n. Then we list all self-dual cyclic codes over of length 28 and 60, respectively.