{"title":"Two classes of new optimal ternary cyclic codes","authors":"Yan Liu, X. Cao, W. Lu","doi":"10.3934/amc.2021033","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\mathcal{C}_{(2,v)} $\\end{document}</tex-math></inline-formula> by using monomials <inline-formula><tex-math id=\"M2\">\\begin{document}$ x^2 $\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M3\">\\begin{document}$ x^v $\\end{document}</tex-math></inline-formula> for some suitable <inline-formula><tex-math id=\"M4\">\\begin{document}$ v $\\end{document}</tex-math></inline-formula> and explain the novelty of the codes. Furthermore, the weight distribution of <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\mathcal{C}_{(2,v)}^{\\perp} $\\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M6\">\\begin{document}$ v = \\frac{3^{m}-1}{2}+2(3^{k}+1) $\\end{document}</tex-math></inline-formula> is determined.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics of Communications","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.3934/amc.2021033","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 9
Abstract
Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes \begin{document}$ \mathcal{C}_{(2,v)} $\end{document} by using monomials \begin{document}$ x^2 $\end{document} and \begin{document}$ x^v $\end{document} for some suitable \begin{document}$ v $\end{document} and explain the novelty of the codes. Furthermore, the weight distribution of \begin{document}$ \mathcal{C}_{(2,v)}^{\perp} $\end{document} for \begin{document}$ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $\end{document} is determined.
Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes \begin{document}$ \mathcal{C}_{(2,v)} $\end{document} by using monomials \begin{document}$ x^2 $\end{document} and \begin{document}$ x^v $\end{document} for some suitable \begin{document}$ v $\end{document} and explain the novelty of the codes. Furthermore, the weight distribution of \begin{document}$ \mathcal{C}_{(2,v)}^{\perp} $\end{document} for \begin{document}$ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $\end{document} is determined.
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.
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