Weierstrass Semigroup and Pure Gaps at several points on the $GK$ curve

Abstract

We determine the Weierstrass semigroup $H(P_{\infty },P_1,\dots ,P_m)$ at several points on the $GK$ curve. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty },P_1,\dots ,P_m)$. Finally, we apply the results to obtain AG codes with good relative parameters.