We explicitly compute the Weierstrass semigroups of specific places, along with a set of defining functions, within an asymptotically optimal tower of function fields up to level eight.

Abstract Let 𝔽 be the finite field of order q2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+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 Hq+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 H72.${{\mathcal{H}}_{72}}.$

Weierstrass semigroups are well known along the literature. We present a new family of non- Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for computing non-Weierstrass semigroups with genus as large as desired.

Abstract Let F be the finite field of order q2. In this paper we continue the study in [24], [23], [22] of F-maximal curves defined by equations of type yn=xℓ(xm+1). ${{y}^{n}}={{x}^{\ell }}\left( {{x}^{m}}+1 \right).$New results are obtained via certain subcovers of the nonsingular model of vN=ut2−u ${{v}^{N}}={{u}^{{{t}^{2}}}}-u$where q = tα, α ≥ 3 is odd and N = (tα + 1)/(t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve.

A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from certain curves defined by equations with separated variables. The recovery of erasures is obtained by means of Lagrangian interpolation in general, and simply by one addition in some particular cases.

We point out a characterization of the Ree curve which involves the number of rational points, the genus, and the shape of two elements of the Weierstrass semigroup at a rational point.

We investigate the structure of the generalized Weierstrass semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to associate them with combinatorial objects, the Poincaré series and the semigroup polynomial. We show that this Poincaré series determines completely the generalized Weierstrass semigroup and it is entirely determined by the semigroup polynomial. We finish the paper by describing the functional equations occurring to the Poincaré series under the hypothesis of a symmetric generalized Weierstrass semigroup.

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.

We study maximal curves arising from Chebyshev polynomials, where in particular some results from Garcia–Stichtenoth [4] are revisited and generalized.

Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper ,we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7\leq q\leq 16$. In particular, by using a result in Kudo and Harashita(2016) paper, the set $\mathbf{M}(7^2)$ is completely determined.

ABSTRACT We characterize certain maximal curves over finite fields whose plane models are of Hurwitz type, namely . We also consider maximal hyperelliptic curves of maximal genus. Finally, we discuss maximal curves of type through class field theory.

Let ng be the number of numerical semigroups of genus g. We present an approach to compute ng by using even gaps, and the question: Is it true that ng+1>ng? is investigated. Let Nγ(g) be the number of numerical semigroups of genus g whose number of even gaps equals γ. We show that Nγ(g)=Nγ(3γ) for γ≤⌊g∕3⌋ and Nγ(g)=0 for γ>⌊2g∕3⌋; thus the question above is true provided that Nγ(g+1)>Nγ(g) for γ=⌊g∕3⌋+1,…,⌊2g∕3⌋. We also show that Nγ(3γ) coincides with fγ, the number introduced by Bras-Amorós (2012) in connection with semigroup-closed sets. Finally, the stronger possibility fγ∼φ2γ arises being φ=(1+5)∕2 the golden number.

We construct examples of curves defined over the finite field Fq6 which are covered by the GK-curve. Thus such curves are maximal over Fq6 although they cannot be covered by the Hermitian curve for q>2. We also give examples of maximal curves that cannot be Galois covered by the Hermitian curve over the finite field Fq2n with n>3 odd and q>2. We point out some applications to codes related to an array coming from telescopic semigroups.

We study Algebraic Geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.

We generalize the concept of near weight stated in Carvalho et al. (IEEE Trans Inf Theory 53(5):1919–1924, 2007) in the sense that we consider maps to arbitrary well-ordered semigroups instead of the nonnegative integers. This concept can be used as a tool to study AG codes based on more than one point via elementary methods only, as well as to construct codes from higher dimensional varieties.

We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.

We investigate complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves which is calculated by using exponential sums via Coulter's approach. We also single out some examples of maximal curves.

We characterize certain maximal curves over finite fields defined by equations of type yn=xm+x. Moreover, we show that a maximal curve over Fq2 defined by the affine equation yn=f(x), where f(x)∈Fq2[x] is separable of degree coprime to n, is such that n is a divisor of q+1 if and only if f(x) has a root in Fq2. In this case, all the roots of f(x) belong to Fq2; cf. Theorems 1.2 and 4.3 in Garcia and Tafazolian (2008) [9].

We investigate one-point algebraic geometry codes defined from curves related to the Hermitian curve. We obtain codes attaining new records on the parameters.

The aim of this paper is to give a characterization of a maximal curve given by the equation x(n) + y(m) = 1 over a finite field F-q2.