Weierstrass Semigroups From a Tower of Function Fields Attaining the Drinfeld-Vladut Bound

For applications in algebraic geometric codes, an explicit description of bases of Riemann-Roch spaces of divisors on function fields over finite fields is needed. We investigate the third function field $F^{(3)}$ in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vl{\u{a}}du{\c{t}} bound. We construct bases for the related Riemann-Roch spaces on $F^{(3)}$ 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 $F^{(3)}$. All of these results can be viewed as a generalization of the previous work done by Voss and H\o{}holdt (1997).