Weierstrass semigroup at $m+1$ rational points in maximal curves which cannot be covered by the Hermitian curve

TL;DR

This paper determines the Weierstrass semigroup at multiple rational points on certain maximal curves not covered by Hermitian curves, and uses this to improve algebraic geometry codes.

Contribution

It provides explicit descriptions of Weierstrass semigroups at multiple points on these special curves and applies this to construct better AG codes.

Findings

Explicit semigroup descriptions for multiple points on these curves

Conditions for identifying pure gaps in the semigroup

Construction of AG codes with improved parameters

Abstract

We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teher\'an-Herrera, and Torres. Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative parameters than comparable one-point AG codes arising from these curves.