Cyclic codes over finite rings
type: Item
instance of: scholarly article
title: Cyclic codes over finite rings
zbMATH Open document ID: 0886.94014
DOI: 10.1016/S0012-365X(97)00006-X
author: Marcus Greferath
published in: Discrete Mathematics
publication date: 1998-03-24T00:00:00Z
review text: A linear left code \(C\) of length \(n\) over a finite ring \(R\) is a submodule of \({}_R R^n\). It is called splitting if it is a direct summand of \({}_R R^n\). \(C\) is a cyclic linear left code if it is a left ideal of \(R[x]/(x^n -1)\) and it is called splitting if it is a direct summand of \({}_R (R[x]/(x^n -1))\). The main result is a complete characterization of cyclic splitting codes as follows: Theorem: For a cyclic linear left code of length \(n\) over a finite ring \(R\) the following are equivalent: (a) \(C\) is a splitting code. (b) There exists a divisor \(g\) of \(x^n -1\) in \(R[x]\) such that \(C=R[x]g/(x^n -1)\).
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 94B15
zbMATH DE Number: 1110121
zbMATH Keywords: cyclic linear codes
zbMATH Keywords: finite rings
zbMATH Keywords: cyclic splitting codes
reviewed by: Ian Blake
MaRDI profile type: Publication
cites work: Modules with decompositions that complement direct summands
cites work: Codes over certain rings
cites work: Codes over integer residue rings
cites work: Modular and \(p\)-adic cyclic codes
cites work: Self-dual codes over the integers modulo 4
cites work: Q4283400
cites work: The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes
cites work: Über die Identität von MacWilliams für die Gewichtsfunktion von Codes
cites work: Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4. (Self-dual codes over the ring of integers modulo 4)
cites work: Q3999650
cites work: Lee metric codes over integer residue rings (Corresp.)
cites work: On BCH codes over arbitrary integer tings (Corresp.)
cites work: Codes over Zm
cites work: Codes over Zm, revisited
cites work: Q4146667
description: scientific article; zbMATH DE number 1110121