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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 X were investigated and the sets of minimal generators were determined for all points in X (F q 2) and X (F q 6) \ X (F q 2) . This paper completes their work by settling the remaining cases, that is, for points in 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 X (F q 7) \ X (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 F q 7 -rational points of the GK-curve. [ABSTRACT FROM AUTHOR]

In this note we investigate minimal linear codes arising from Hermitian varieties and quadrics. We study their parameters and formulate some open problems about their weight distribution. [ABSTRACT FROM AUTHOR]

Let C be an irreducible plane curve of PG (2 , K) where K is an algebraically closed field of characteristic p ≥ 0. A point Q ∈ C is an inner Galois point for C if the projection π Q from Q is Galois. Assume that C has two different inner Galois points Q 1 and Q 2 , both simple. Let G 1 and G 2 be the respective Galois groups. Under the assumption that G i fixes Q i , for i = 1 , 2 , we provide a complete classification of G = 〈 G 1 , G 2 〉 and we exhibit a curve for each such G. Our proof relies on deeper results from group theory. [ABSTRACT FROM AUTHOR]

The complete classification of the orbits on subspaces under the action of the projective stabilizer of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of PGL (2 , q 2) (and PSL (2 , q 2)) arising from the Hermitian Veronese curve in PG (3 , q 2) , a maximal rational curve embedded on a smooth Hermitian surface with some fascinating properties. The study of its orbits leads to a new construction of quasi-Hermitian surfaces: sets of points with the same combinatorial and geometric properties as a non-degenerate Hermitian surface. [ABSTRACT FROM AUTHOR]

In this paper, X is an algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of positive characteristic p , G is an automorphism group of X which fixes K element-wise, and, for a point P ∈ X , G P is the subgroup of G which fixes P. The question "how large G P can be compared to g " has been the subject of several papers. We are concerned with the case where the second ramification group G P (2) of G P is trivial. Under this condition Theorem 3.1 states that if | G P | > 12 (g − 1) then X is either an ordinary hyperelliptic curve, or it has zero p -rank and p ≠ 3. More precisely, up to birational equivalence, there exists a separable p -linearized polynomial L (T) ∈ K T of degree q such that an affine equation of X is L (y) = a x + 1 / x with a ∈ K ⁎ in the former case, and L (y) = x 3 + b x with b ∈ K in the latter case. In 1987 Nakajima proved that if X is an ordinary curve (more generally, the second ramification group of G is trivial for every P ∈ X), then the order of G does not exceed 84 g (g − 1). We show that Theorem 3.1 together with some refinements of Nakajima's computations provide a slight improvement in Nakajima's bound from 84 g (g − 1) to 48 (g − 1) 2. [ABSTRACT FROM AUTHOR]

In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X6+Y6+ℨ6+(X2+Y2+ℨ2)(X4+Y4+ℨ4)−12X2Y2ℨ2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman's 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192. [ABSTRACT FROM AUTHOR]