{"title":"$ \\mathbb{Z}_{p^r}\\mathbb{Z}_{p^s}\\mathbb{Z}_{p^t} $-additive cyclic codes","authors":"Raziyeh Molaei, K. Khashyarmanesh","doi":"10.3934/amc.2022079","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M3\">\\begin{document}$ p $\\end{document}</tex-math></inline-formula> be a prime number and <inline-formula><tex-math id=\"M4\">\\begin{document}$ r, s, t $\\end{document}</tex-math></inline-formula> be positive integers such that <inline-formula><tex-math id=\"M5\">\\begin{document}$ r\\le s\\le t $\\end{document}</tex-math></inline-formula>. A <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\mathbb{Z}_{p^r}\\mathbb{Z}_{p^s}\\mathbb{Z}_{p^t} $\\end{document}</tex-math></inline-formula>-additive code is a <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\mathbb{Z}_{p^t} $\\end{document}</tex-math></inline-formula>-submodule of <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\mathbb{Z}_{p^r}^{\\alpha} \\times \\mathbb{Z}_{p^s}^{\\beta} \\times \\mathbb{Z}_{p^t}^{\\gamma} $\\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M9\">\\begin{document}$ \\alpha, \\beta, \\gamma $\\end{document}</tex-math></inline-formula> are positive integers. In this paper, we study <inline-formula><tex-math id=\"M10\">\\begin{document}$ \\mathbb{Z}_{p^r}\\mathbb{Z}_{p^s}\\mathbb{Z}_{p^t} $\\end{document}</tex-math></inline-formula>-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring <inline-formula><tex-math id=\"M11\">\\begin{document}$ R = \\mathbb{Z}_{p^r}[x]/\\big<x^\\alpha-1\\big> \\times \\mathbb{Z}_{p^s}[x]/\\big<x^\\beta-1\\big> \\times \\mathbb{Z}_{p^t}[x]/\\big<x^\\gamma-1\\big> $\\end{document}</tex-math></inline-formula>. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"11 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics of Communications","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.3934/amc.2022079","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 1
Abstract
Let \begin{document}$ p $\end{document} be a prime number and \begin{document}$ r, s, t $\end{document} be positive integers such that \begin{document}$ r\le s\le t $\end{document}. A \begin{document}$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $\end{document}-additive code is a \begin{document}$ \mathbb{Z}_{p^t} $\end{document}-submodule of \begin{document}$ \mathbb{Z}_{p^r}^{\alpha} \times \mathbb{Z}_{p^s}^{\beta} \times \mathbb{Z}_{p^t}^{\gamma} $\end{document}, where \begin{document}$ \alpha, \beta, \gamma $\end{document} are positive integers. In this paper, we study \begin{document}$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $\end{document}-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring \begin{document}$ R = \mathbb{Z}_{p^r}[x]/\big \times \mathbb{Z}_{p^s}[x]/\big \times \mathbb{Z}_{p^t}[x]/\big $\end{document}. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.
Let \begin{document}$ p $\end{document} be a prime number and \begin{document}$ r, s, t $\end{document} be positive integers such that \begin{document}$ r\le s\le t $\end{document}. A \begin{document}$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $\end{document}-additive code is a \begin{document}$ \mathbb{Z}_{p^t} $\end{document}-submodule of \begin{document}$ \mathbb{Z}_{p^r}^{\alpha} \times \mathbb{Z}_{p^s}^{\beta} \times \mathbb{Z}_{p^t}^{\gamma} $\end{document}, where \begin{document}$ \alpha, \beta, \gamma $\end{document} are positive integers. In this paper, we study \begin{document}$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $\end{document}-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring \begin{document}$ R = \mathbb{Z}_{p^r}[x]/\big \times \mathbb{Z}_{p^s}[x]/\big \times \mathbb{Z}_{p^t}[x]/\big $\end{document}. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.
期刊介绍:
Advances in Mathematics of Communications (AMC) publishes original research papers of the highest quality in all areas of mathematics and computer science which are relevant to applications in communications technology. For this reason, submissions from many areas of mathematics are invited, provided these show a high level of originality, new techniques, an innovative approach, novel methodologies, or otherwise a high level of depth and sophistication. Any work that does not conform to these standards will be rejected.
Areas covered include coding theory, cryptology, combinatorics, finite geometry, algebra and number theory, but are not restricted to these. This journal also aims to cover the algorithmic and computational aspects of these disciplines. Hence, all mathematics and computer science contributions of appropriate depth and relevance to the above mentioned applications in communications technology are welcome.
More detailed indication of the journal''s scope is given by the subject interests of the members of the board of editors.