Related papers: Weierstrass semigroups and the order bound
In this paper we treat several topics regarding numerical Weierstrass semigroups and the theory of Algebraic Geometric Codes associated to a pair $(X, P)$, where $X$ is a projective curve defined over the algebraic closure of the finite…
In this paper we compute the order (or Feng-Rao) bound on the minimum distance of one-point algebraic geometry codes, when the Weierstrass semigroup at the point Q is an Arf semigroup. The results developed to that purpose also provide the…
We present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at infinity of a singular plane model for the curve. As a byproduct, the method…
Garcia and Stichtenoth discovered two towers of function fields that meet the Drinfeld-Vl\u{a}du\c{t} bound on the ratio of the number of points to the genus. For one of these towers, Garcia, Pellikaan and Torres derived a recursive…
We study suitable parameters and relations in a numerical semigroup S. When S is the Weierstrass semigroup at a rational point P of a projective curve C, we evaluate the Feng-Rao order bound of the associated family of Goppa codes. Further…
A numerical semigroup is a subset of N containing 0, closed under addition and with finite complement in N. An important example of numerical semigroup is given by the Weierstrass semigroup at one point of a curve. In the theory of…
The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behaviour of such bounds for r > 1…
In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the…
In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all $\mathbb F_{q^2}$-rational points of the curve is determined; the Feng-Rao designed minimum distance is…
In this work, we are concerned with the structure of sparse semigroups and some applications of them to Weierstrass points. We manage to describe, classify and find an upper bound for the genus of sparse semigroups. We also study the…
We describe the second (generalized) Feng-Rao distance for elements in an Arf numerical semigroup that are greater than or equal to the conductor of the semigroup. This provides a lower bound for the second Hamming weight for one point AG…
In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Suzuki curve $\mathcal{S}_q$. As the point $P$ varies, exactly two possibilities arise for $H(P)$: one for the…
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 investigate the third function field $ F^{(3)} $ in a tower of…
We continue the investigation of curves of type p,q started in [KKW]. We study the space of such curves and the space of nodal curves with prescribed Weierstra\ss semigroup. A necessary and sufficient criterion for a numerical semigroup to…
We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes are consequences of…
The aim of this paper is to review the main techniques in the computation of Weierstra\ss semigroup at several points of curves defined over perfect fields, with special emphasis on the case of two points. Some hints about the usage of some…
We determine de Weierstrass semigroup of a pair of certain rational points on the GK-curves. We use this semigroup to obtain two-point AG codes with better parameters than comparable one-point AG codes arising from these curves. These…
The most successful method to obtain lower bounds for the minimum distance of an algebraic geometric code is the order bound, which generalizes the Feng-Rao bound. We provide a significant extension of the bound that improves the order…
A numerical semigroup is said to be Weierstrass if it is the semigroup of pole orders of rational functions that are regular at all but one point of some compact Riemann surface or smooth algebraic curve. Hurwitz asked in 1892 whether all…
The second Feng-Rao number of every inductive numerical semigroup is explicitly computed. This number determines the asymptotical behaviour of the order bound for the second Hamming weight of one-point AG codes. In particular, this result…