{"title":"Weierstrass semigroups on the third function field in a tower attaining the Drinfeld-Vlăduţ bound","authors":"Shudi Yang, Chuangqiang Hu","doi":"10.3934/amc.2022066","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>For applications in algebraic geometry codes, an explicit description of bases of the Riemann-Roch spaces over function fields is needed. We investigate the third function field <inline-formula><tex-math id=\"M1\">\\begin{document}$ F^{(3)} $\\end{document}</tex-math></inline-formula> in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vlăduţ bound. We construct new bases for the related Riemann-Roch spaces of <inline-formula><tex-math id=\"M2\">\\begin{document}$ F^{(3)} $\\end{document}</tex-math></inline-formula> and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on <inline-formula><tex-math id=\"M3\">\\begin{document}$ F^{(3)} $\\end{document}</tex-math></inline-formula>. All of these results can be viewed as a generalization of the previous work done by Voss and Høholdt (1997).</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"6 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics of Communications","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.3934/amc.2022066","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 1
Abstract
For applications in algebraic geometry codes, an explicit description of bases of the Riemann-Roch spaces over function fields is needed. We investigate the third function field \begin{document}$ F^{(3)} $\end{document} in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vlăduţ bound. We construct new bases for the related Riemann-Roch spaces of \begin{document}$ F^{(3)} $\end{document} and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on \begin{document}$ F^{(3)} $\end{document}. All of these results can be viewed as a generalization of the previous work done by Voss and Høholdt (1997).
For applications in algebraic geometry codes, an explicit description of bases of the Riemann-Roch spaces over function fields is needed. We investigate the third function field \begin{document}$ F^{(3)} $\end{document} in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vlăduţ bound. We construct new bases for the related Riemann-Roch spaces of \begin{document}$ F^{(3)} $\end{document} and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on \begin{document}$ F^{(3)} $\end{document}. All of these results can be viewed as a generalization of the previous work done by Voss and Høholdt (1997).
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.