On Abelian codes over $${\mathbb {Z}}_{m}$$ Z m

Abstract

In this paper, we describe the Chinese Remainder Theorem for studying Abelian codes of length \(N\) over the ring \({\mathbb {Z}}_{m}\), where \(m=\prod _{i=1}^{s}p_{i}, \)\(k=\prod _{i=1}^{s}p_{i}^{t_{i}}, \, N=kn, \, p_{i}\) are distinct primes, \(s\) is a positive integer, \(t_{i}\) are positive integers and \(n\) is a positive integer prime to \(k.\) The structure theorems for Abelian codes and their duals in \({\mathbb {Z}}_{m}G\) are obtained, where \(G=C_{k} \times H,\)\(C_{k}\) denotes the cyclic group of order \(k\) and \(H\) denotes a group of order \(n.\) The existence of self-orthogonal and the nonexistence of self-dual Abelian codes over \({\mathbb {Z}}_{m}\) are studied.