Related papers: Weierstrass semigroups and the order bound
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 calculating…
In this work we determine the so-called minimal generating set of the Weierstrass semigroup of certain $m$ points on curves $\mathcal{X}$ with plane model of the type $f(y) = g(x)$ over $\mathbb{F}_{q}$, where $f(T),g(T)\in…
We determine the Weierstrass semigroup $H(P_{\infty}, P_{1}, \ldots , P_{m})$ at several points on the $GK$ curve. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty}, P_{1}, \ldots , P_{m})$. Finally, we…
Sensing and imaging are among the most important applications of quantum information science. To investigate their fundamental limits and the possibility of quantum enhancements, researchers have for decades relied on the quantum…
In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchm\'aros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the…
We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the…
We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\alpha_i)^{\lambda_i}$ over $K$, the algebraic closure of $\mathbb{F}_q$, where…
We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teher\'an-Herrera, and Torres. Furthermore, we…
We improve previously known lower bounds for the minimum distance of certain two-point AG codes constructed using a Generalized Giulietti-Korchmaros curve (GGK). Castellanos and Tizziotti recently described such bounds for two-point codes…
Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the St\"orh-Voloch theory and sets new bounds to the number of…
Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandi\'c, Kramar-Fijav\v{z}, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of…
Weil's theorem gives the most standard bound on the number of points of a curve over a finite field. This bound was improved by Ihara and Oesterl\'e for larger genus. Recently, Hallouin and Perret gave a new point of view on these bounds,…
The Weierstrass semigroups and pure gaps can be helpful in constructing codes with better parameters. In this paper, we investigate explicitly the minimal generating set of the Weierstrass semigroups associated with several totally ramified…
We study plane curves of type p,q having only nodes as singularities. Every Weierstra\ss semigroup is the Weierstra\ss semigroup of such a curve at its place at infinity for properly chosen p,q. We construct plane curves of type p,q with…
Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal…
We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety $M^S_{g,1}$ of $M_{g,1}$ consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible…
For applications in algebraic geometric codes, an explicit description of bases of Riemann-Roch spaces of divisors on function fields over finite fields is needed. We give an algorithm to compute such bases for one point divisors, and…
In this paper we investigate two-point algebraic-geometry codes (AG codes) coming from the Beelen-Montanucci (BM) maximal curve. We study properties of certain two-point Weierstrass semigroups of the curve and use them for determining a…
The Clifford defect is a rational number associated to the Weierstrass semigroup at a given point of an algebraic curve. It describes the error-correcting capability of the so-called Modified Algorithm for decoding the corresponding…
We use a version of the Trotter-Kato approximation theorem for strongly continuous semigroups in order to study flows on growing networks. For that reason we use the abstract notion of direct limits in the sense of category theory.