AbstractIn 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 $${\mathcal {X}}$$
X
were investigated and the sets of minimal generators were determined for all points in $${\mathcal {X}}(\mathbb {F}_{q^2})$$
X
(
F
q
2
)
and $${\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})$$
X
(
F
q
6
)
\
X
(
F
q
2
)
. This paper completes their work by settling the remaining cases, that is, for points in $${\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})$$
X
(
F
¯
q
)
\
X
(
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 $${\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})$$
X
(
F
q
7
)
\
X
(
F
q
)
and we give a bound on the Feng–Rao minimum distance $$d_{ORD}$$
d
ORD
. For $$q=3$$
q
=
3
we provide a table that also reports the exact values of $$d_{ORD}$$
d
ORD
. As a further application we construct quantum codes from $$\mathbb {F}_{q^7}$$
F
q
7
-rational points of the GK-curve.
AG codes from $${{\mathbb{F}}_{q^7}}$$-rational points of the GK maximal curve