Polycyclic codes as invariant subspaces
type: Item
instance of: scholarly article
title: Polycyclic codes as invariant subspaces
zbMATH Open document ID: 1465.94126
author: Minjia Shi
author: Zahra Sepasdar
author: Patrick Solé
author: Xiao-Xiao Li
published in: Finite Fields and their Applications
publication date: 2021-02-18T00:00:00Z
review text: Polycyclic codes of length \(n\) over a finite field \(\mathbb{F}\) can be described as ideals in the quotient ring of \(\mathbb{F}[x]/(f),\) where \(f\) is some polynomial of degree \(n.\) The main idea in this work is to consider the codes in the class as invariant subspaces under the action of an endomorphism which can allow some advanced techniques from linear algebra to help solving the problem. The authors apply a polyshift transform that admits for a matrix the companion matrix of the polynomial \(f(x)\). A notion of minimal invariant subspace is introduced, driven by the factorization of \(f(x)\) into irreducible factors over the base field. This allows to describe any invariant subspace as a direct sum of its intersections with these spaces. Similarly, an analogue of the idempotent of cyclic codes is developed and a new bound on the minimum distance is derived which improves, in some cases, the previously established BCH-like bound. The paper concludes with some open problems.
reviewed by: Nikolay Yankov
Mathematics Subject Classification ID: 94B15
Mathematics Subject Classification ID: 11A15
zbMATH DE Number: 7312736
zbMATH Keywords: polycyclic codes
zbMATH Keywords: cyclic codes
zbMATH Keywords: BCH bound
zbMATH Keywords: invariant subspaces
MaRDI profile type: Publication
full work available at URL: j.ffa.2020.101760
OpenAlex ID: W3088890156
cites work: On the duality and the direction of polycyclic codes
cites work: Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes
cites work: On the additive cyclic structure of quasi-cyclic codes
cites work: Q5614003
cites work: Pseudo-cyclic Codes and the Construction of Quantum MDS Codes
cites work: Dual generalizations of the concept of cyclicity of codes
cites work: Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes
cites work: On polycyclic codes over a finite chain ring
cites work: On repeated-root multivariable codes over a finite chain ring
cites work: Q5665102
cites work: Constacyclic codes as invariant subspaces
cites work: Quasi-twisted codes with constacyclic constituent codes
cites work: A modified Gilbert-Varshamov bound for self-dual quasi-twisted codes of index four
DOI: 10.1016/J.FFA.2020.101760
Recommended article: Q4993763
Recommended article: Constacyclic codes as invariant subspaces
Recommended article: Subspace Polynomials and Cyclic Subspace Codes
Recommended article: Cyclic subspace codes via subspace polynomials
Recommended article: A characterization of cyclic subspace codes via subspace polynomials
Recommended article: Monomial codes seen as invariant subspaces
Recommended article: Q4993764
Recommended article: Polynomial invariants for binary linear codes
Recommended article: Polyadic Constacyclic Codes
Recommended article: On polycyclic codes over a finite chain ring
description: scientific article; zbMATH DE number 7312736