In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve \({\mathcal {X}}\) were investigated and the sets of minimal generators were determined for all points in \({\mathcal {X}}(\mathbb {F}_{q^2})\) and \({\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})\). This paper completes their work by settling the remaining cases, that is, for points in \({\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})\). As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in \({\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})\) and we give a bound on the Feng–Rao minimum distance \(d_{ORD}\). For \(q=3\) we provide a table that also reports the exact values of \(d_{ORD}\). As a further application we construct quantum codes from \(\mathbb {F}_{q^7}\)-rational points of the GK-curve.

在 Beelen 和 Montanucci（Finite Fields Appl 52:10–29, 2018）以及 Giulietti 和 Korchmáros（Math Ann 343:229–245, 2009）中，Giulietti–Korchmáros 曲线上的 Weierstrass 半群{X} }\)进行了研究，并为\({\mathcal {X}}(\mathbb {F}_{q^2})\)和\({\mathcal {X} }(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})\)。本文通过解决剩余的情况来完成他们的工作，即对于\({\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X} }}( \mathbb {F}_{q^6})\)。作为对 AG 码的应用，我们从一个点确定单点码的对偶的维数和长度。\({\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})\)并且我们给出限制在 Feng-Rao 最小距离\(d_{ORD}\) 上。对于\(q=3\)，我们提供了一个表格，其中还报告了\(d_{ORD}\)的确切值。作为进一步的应用，我们从GK 曲线的\(\mathbb {F}_{q^7}\) -有理点构造量子代码。