Constacyclic codes as invariant subspaces
type: Item
instance of: scholarly article
title: Constacyclic codes as invariant subspaces
zbMATH Open document ID: 1160.94014
author: A. J. van Zanten
author: Diana Radkova
published in: Linear Algebra and its Applications
publication date: 2008-12-12T00:00:00Z
review text: A constacyclic code (sometimes also referred to as pseudo-cyclic code) is a linear code over a field GF\((q)\) invariant under the permutation \(\psi_a:\) \((c_0,c_1\ldots,c_{n-1})\mapsto (ac_{n-1},c_0,c_1\ldots,c_{n-1})\) (for a fixed field element \(a\)). (In the introduction of the article the definition is not correct). The matrix \(A(n,a)\) of \(\psi_a\) (with respect to the standard basis) fullfils the equation \(A(n,a)^{-1}=A(n,a^{-1})^t\) (here, there is a misprint). In this article, generalizations of results known for cyclic codes are derived for constacyclic codes \(\mathcal C\), partly by writing \(\mathcal C\) as direct sum of some mini\-mal \(\psi_a-\)invariant subspaces \(U_i\) and using linear algebra. For instance there are defined primitive idempotent matrices fixing an \(U_i\), anihilating the other \(U_j\), and having as sum a matrix fixing exactly \(\mathcal C\). Furthermore, the authors prove a generalization of the BCH-bound and of the Hartmann-Tzeng-Roos bound [see \textit{C. Roos}, J. Comb. Theory, Ser. A 33, 229--232 (1982; Zbl 0497.94011)] for the minimal distance of constacyclic codes and apply this bound to an example of a constacyclic code of parameters \(n=25\), \(q=7\), \(a=-1\) showing that it has minimum distance \(\geq 6\).
reviewed by: Ralph-Hardo Schulz
Mathematics Subject Classification ID: 94B15
Mathematics Subject Classification ID: 94B60
Mathematics Subject Classification ID: 94B65
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 5382426
zbMATH Keywords: constacyclic codes
zbMATH Keywords: cyclic codes
zbMATH Keywords: pseudo-cyclic codes
zbMATH Keywords: invariant subspaces
zbMATH Keywords: BCH bound
zbMATH Keywords: Hartmann-Tzeng-Roos bound
MaRDI profile type: Publication
OpenAlex ID: W1979614101
cites work: Q4146667
cites work: Q5598073
cites work: Q4822436
cites work: A generalization of the BCH bound for cyclic codes, including the Hartmann-Tzeng bound
cites work: A new lower bound for the minimum distance of a cyclic code
DOI: 10.1016/J.LAA.2008.09.036
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description: scientific article; zbMATH DE number 5382426