Weierstrass Semigroup and Pure Gaps at Several Points on the GK Curve

Abstract

Abstract. We determine the Weierstrass semigroup H(P ∞ , P 1 , . . . , 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 ∞ , P 1 , . . . , P m ). Finally, we apply the results to obtain AG codes with good relative parameters.

Generalized Weierstrass semigroups at several points on certain maximal curves which cannot be covered by the Hermitian curve

“…Many applications of these curves in coding theory have been made in recent years, see e.g. [1], [2], [8], [12], [19] and [31]. Another generalization of the GK curve over F q 2n (n odd) has been introduced by Beelen and Montanucci in [4], which is now known as BM curves.…”

Pure gaps on curves with many rational places

“…For those curves with affine equation given by y m = f (x) λ where m ≥ 2, λ ≥ 1 and f (x) is a separable polynomial over F q , general results on gaps and pure gaps can be found in [1,5,18,29]. Applications to codes on particular curves such as the Giulietti-Korchmáros curve, the Garcia-Güneri-Stichtenoth curve, and quotients of the Hermitian curve can be found in [19,28,31]. In [1] the authors use a decomposition of certain Riemann-Roch vector spaces due to Maharaj (Theorem 2.2) to describe gaps at one place arithmetically.…”