AG codes from $${{\mathbb{F}}_{q^7}}$$-rational points of the GK maximal curve

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 $${\mathcal {X}}$$ 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.

𝔽 p 2 -maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Let 𝔽 be the finite field of order q 2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curve H q + 1 : y q + 1 = x q + x ${{\mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$ is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $\mathcal{x}$ of genus g ≥ 2 with | Aut(𝒳) | > 84(g − 1) is Galois-covered by H q + 1 . ${{\mathcal{H}}_{q+1}}.$ The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $\mathcal{x}$ for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H 72 . ${{\mathcal{H}}_{72}}.$

On space maximal curves

Any maximal curve X is equipped with an intrinsic embedding π: X → Pr which reveal outstanding properties of the curve. By dealing with the contact divisors of the curve π(X) and tangent lines, in this paper we investigate the first positive element that the Weierstrass semigroup at rational points can have whenever r = 3 and π(X) is contained in a cubic surface.

ON THE DIMENSION OF SPACES OF ALGEBRAIC CURVES PASSING THROUGH $ n $-INDEPENDENT NODES

Let the set of nodes $ \LARGE{x} $ in the plain be $ n $-independent, i.e., each node has a fundamental polynomial of degree $ n $. Suppose also that $ \vert \LARGE{x} \normalsize \vert \mathclose{=} (n \mathclose{+} 1) \mathclose{+} n \mathclose{+} \cdots \mathclose{+} (n \mathclose{-} k \mathclose{+} 4) \mathclose{+} 2 $ and $ 3 \mathclose{\leq} k \mathclose{\leq} n \mathclose{-} 1 $. We prove that there can be at most 4 linearly independent curves of degree less than or equal to $ k $ passing through all the nodes of $ \LARGE{x} $. We provide a characterization of the case when there are exactly 4 such curves. Namely, we prove that then the set $ \LARGE{x} $ has a very special construction: all its nodes but two belong to a (maximal) curve of degree $ k \mathclose{-} 2 $. At the end, an important application to the Gasca-Maeztu conjecture is provided.