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

TL;DR

This paper explicitly constructs bases for Riemann-Roch spaces and calculates Weierstrass semigroups on a specific function field tower reaching the Drinfeld-Vladut bound, aiding algebraic geometric code design.

Contribution

It provides explicit bases and semigroup calculations for a third function field in a tower attaining the Drinfeld-Vladut bound, extending previous work.

Findings

Explicit bases for Riemann-Roch spaces constructed

Weierstrass semigroups and pure gaps calculated

Generalizes previous results by Voss and Høholdt

Abstract

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).