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

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).