The concatenated structure of cyclic codes over $\mathbb {Z}_{p^2}$

Abstract

Let $N=p^kn$ where p is a prime, and $k,\,n$ are positive integers satisfying $\mathrm{gcd}(p,\,n)=1.$ We present a canonical form decomposition for every cyclic code over $\mathbb {Z}_{p^2}$ of length N,  where each subcode is concatenated by a basic irreducible cyclic code over $\mathbb {Z}_{p^2}$ of length n as the inner code and a constacyclic code over a Galois extension ring of $\mathbb {Z}_{p^2}$ of length $p^k$ as the outer code. By determining their outer codes, we present a precise description for cyclic codes over $\mathbb {Z}_{p^2}$ when $p\ne 2,$ give precisely dual codes and investigate self-duality for cyclic codes over $\mathbb {Z}_{p^2}.$ We end by listing cyclic self-dual codes over $\mathbb {Z}_9$ of length 33.