Weierstrass Semigroup, Pure Gaps and Codes on Function Fields

Abstract

We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^m={\prod }_{i=1}^r(x-{\alpha }_i{)}^{{\lambda }_i}$ over $K$, the algebraic closure of ${\mathbb{F}}_q$, where ${\alpha }_1,\dots ,{\alpha }_r\in K$ are pairwise distinct elements, and $\gcd (m,{\sum }_{i=1}^r{\lambda }_i)=1$. For an arbitrary function field, from the knowledge of the minimal generating set of the Weierstrass semigroup at two rational places, the set of pure gaps is characterized. We apply these results to construct algebraic geometry codes over certain function fields with many rational places.

arXiv comment24 pages

References in corpus (8)

• A new family of maximal curves
• Hermitian codes from higher degree places
• Weierstrass Semigroups from Kummer Extensions
• One- and Two-Point Codes over Kummer Extensions
• Multi-point Codes over Kummer Extensions
• Pure gaps on curves with many rational places
• Construction of sequences with high nonlinear complexity from a generalization of the Hermitian function field
• On cubic Kummer towers of Garcia, Stichtenoth and Thomas type