On self-dual skew cyclic codes of length \(p^s\) over \(\mathbb{F}_{p^m} + u \mathbb{F}_{p^m} \)
type: Item
instance of: scholarly article
title: On self-dual skew cyclic codes of length \(p^s\) over \(\mathbb{F}_{p^m} + u \mathbb{F}_{p^m} \)
zbMATH Open document ID: 1485.94165
author: Roghayeh Mohammadi Hesari
author: Rashid Rezaei
author: Karim Samei
published in: Discrete Mathematics
publication date: 2021-09-06T00:00:00Z
review text: For a finite field \(\mathbb{F}_{q},\) and \(\mathcal{R}_2 = {\mathbb{F}}_{p^m} + u{\mathbb{F}}_{p^m},\) where \(u^2 = 0,\) a skew-cyclic code is a left ideal of the non-commutative ring \(\mathbb{F}_{q}[x, \theta],\) where and \({\theta}\) is an automorphism of \(\mathbb{F}_{q}.\) Skew cyclic codes of length \(p^s\) over \(\mathcal{R}_2\) are divided into four types: Type 1 (trivial ideals), Type 2 (principal left ideals with non-monic polynomial generators), Type 3 (principal left ideals with monic polynomial generators) and Type 4 (non-principal left ideals). Any polynomial \(f(x)\in\mathcal{R}_2\) can be uniquely expressed as \(f(x)=\sum\limits_{i=0}^{p^s-1}{a_{0i}(x-1)^i}+u\sum\limits_{i=0}^{p^s-1}{a_{1i}(x-1)^i},\) where \(a_{0i}, a_{1i}\in\mathbb{F}_{p^m}.\) The main theme of this work are the special skew cyclic codes such that \(a(x),a_1(x),a_2(x)\in\{(x-1)^j : 0\leq j\leq ps-1\}.\) For \(a(x)\) -- a monic factor of \(x^{p^s}-1\) the sum \((x-1)^na\) is calculated for a skew cyclic code of Type 3 (principal left ideals with monic polynomial generators) \(C =\mathcal{R}_2((x-1)^i +uh(x)(x-1)^t)\) and some additional properties are also shown. For all four types, the number of codewords \(n_C\) of \(C\) is determined as well as the dual of these codes and the self-dual among them are obtained. Some examples of self-dual skew cyclic codes of length \(p^s\) over \(\mathcal{R}_2\) are shown.
reviewed by: Nikolay Yankov
Mathematics Subject Classification ID: 94B15
zbMATH DE Number: 7391497
zbMATH Keywords: skew cyclic code
zbMATH Keywords: self-dual code
zbMATH Keywords: chain ring
MaRDI profile type: Publication
full work available at URL: j.disc.2021.112569
OpenAlex ID: W3193797831
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DOI: 10.1016/J.DISC.2021.112569
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description: scientific article; zbMATH DE number 7391497