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

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.