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Published in:
01-02-2016 | Original Research
On Abelian codes over \({\mathbb {Z}}_{m}\)
Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2016
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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.