The concatenated structure of cyclic codes over

Let where p is a prime, and are positive integers satisfying 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 of length as the outer code. By determining their outer codes, we present a precise description for cyclic codes over when give precisely dual codes and investigate self-duality for cyclic codes over We end by listing cyclic self-dual codes over of length 33.