On the structure of linear and cyclic codes over a finite chain ring
type: Item
instance of: scholarly article
title: On the structure of linear and cyclic codes over a finite chain ring
zbMATH Open document ID: 0963.94042
DOI: 10.1007/PL00012382
author: G. H. Norton
author: Ana Sălăgean
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2000-08-08T00:00:00Z
review text: In [\textit{A. R. Calderbank} and \textit{N. J. A. Sloane}, Des. Codes Cryptography 6, 21-35 (1995; Zbl 0848.94020)], structure theorems are given for linear and cyclic codes over \(\mathbb{Z}_{p^a}\). Such codes received much attention after the discovery that important families of binary nonlinear codes can be obtained as images under a Gray mapping of \(\mathbb{Z}_4\)-linear codes [\textit{A. R. Hammons jun., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane} and \textit{P. Solé}, IEEE Trans. Inf. Theory 40, 301-319 (1994; Zbl 0811.94039)]. The present paper generalizes the results of Calderbank-Sloane (loc. cit.) to a finite chain ring. New proofs are given that avoid the nontrivial commutative algebra invoked in Calderbank-Sloane (loc. cit.). In a follow-up paper [\textit{G. H. Norton} and \textit{A. Sălăgean}, IEEE Trans. Inf. Theory 46, 1060-1067 (2000; Zbl 0963.94043)], the authors use the structure theorems from this paper to obtain results on the minimum Hamming distance of codes over a finite chain ring.
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 1486224
zbMATH Keywords: Galois ring
zbMATH Keywords: linear code
zbMATH Keywords: cyclic code
zbMATH Keywords: minimum Hamming distance
zbMATH Keywords: finite chain ring
reviewed by: L. M. G. M. Tolhuizen
MaRDI profile type: Publication
full work available at URL: pl00012382
OpenAlex ID: W2085147719
description: scientific article; zbMATH DE number 1486224