Weierstrass semigroups on the third function field in a tower attaining the Drinfeld-Vlăduţ bound

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>