New quantum codes from constacyclic codes over the ring \(R_{k,m}\)
type: Item
instance of: scholarly article
title: New quantum codes from constacyclic codes over the ring \(R_{k,m}\)
zbMATH Open document ID: 1484.94032
author: Habibul Islam
author: Om Prakash
author: Ram Krishna Verma
published in: Advances in Mathematics of Communications
publication date: 2022-01-24T00:00:00Z
review text: For positive integers \(k,m\) we define the ring \(R_{k,m}=\mathbb{F}_{p^m}[u_1,\ldots,u_k]/\langle u_i^2-1,u_iu_j-u_ju_i\rangle\) and it is a commutative ring of characteristics \(p\) and order \(p^{2^km}.\) For given element \(\gamma\in R_{k,m}^\ast\) a linear code \(C\) is \(\gamma\)-constacyclic if \(\tau_\gamma(c)=(\gamma c_{n-1},c_0,\ldots,c_{n-2})\in C\) for \(c=(c_0,\ldots,c_{n-1})\in C.\) The authors study the constacyclic codes over new and different non-chain rings in an attempt to construct more new quantum codes. In contrast to others, the authors have used here the Gray map \(\psi\) as the multiplication of a vector by an invertible matrix of order \(2^k\) which has the advantage to enhance the code parameters (particularly, dimension, and minimum distance, etc.) over the parameters obtained by the simple canonical Gray map. The structural properties of constacyclic codes over the ring \(R_{k,m}\) are shown. These codes are later used to obtain new \(q\)-ary quantum codes by employing the known structure of \(\gamma\)-constacyclic codes.
reviewed by: Nikolay Yankov
Mathematics Subject Classification ID: 94B15
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 94B60
zbMATH DE Number: 7461985
zbMATH Keywords: constacyclic code
zbMATH Keywords: gray map
zbMATH Keywords: self-dual code
zbMATH Keywords: quantum code
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: amc.2020097
OpenAlex ID: W3046150819
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DOI: 10.3934/AMC.2020097
description: scientific article; zbMATH DE number 7461985