Bounding the number of points on a curve using a generalization of Weierstrass semigroups

Beelen, Peter; Ruano, Diego

doi:10.1007/s10623-012-9685-3

Cited by 9 publications

(7 citation statements)

References 10 publications

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“…The obvious advantadge of Lewittes' bound with respect to the Geil-Matsumoto bound is that Lewittes' bound is very simple to compute. Furthermore, the results by Beelen and Ruano in [2] allow bounding the number of rational places with nonzero coordinates by #(Λ \ ∪ λ∈Λ\{0} ((q − 1)λ i + Λ)) + 1. It is proved in [8] that the Geil-Matsumoto bound and the Lewittes' bound coincide if and only if qx − qm ∈ Λ for all x ∈ Λ \ {0}, where m is the multiplicity of Λ.…”

Section: The Geil-matsumoto Boundmentioning

confidence: 99%

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Nonhomogeneous Patterns on Numerical Semigroups

Bras-Amorós

García-Sánchez

Vico-Oton

2013

Int. J. Algebra Comput.

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Patterns on numerical semigroups are multivariate linear polynomials, and they are said to admit a numerical semigroup if evaluating the pattern at any nonincreasing sequence of elements of the semigroup gives integers belonging to the semigroup. In a first approach, only homogeneous patterns were analyzed. In this contribution we study conditions for a nonhomogeneous pattern to admit a nontrivial numerical semigroup, and particularize this study to the case the independent term of the pattern is a multiple of the multiplicity of the semigroup. Moreover, for the so-called strongly admissible patterns, the set of numerical semigroups admitting these patterns with fixed multiplicity m forms an m-variety, which allows us to represent this set in a tree and to describe minimal sets of generators of the semigroups in the variety with respect to the pattern. Furthermore, we characterize strongly admissible patterns having a finite associated tree.

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Section: The Geil-matsumoto Boundmentioning

confidence: 99%

Section: Motivating Examplesmentioning

confidence: 99%

Nonhomogeneous Patterns on Numerical Semigroups

Bras-Amorós

García-Sánchez

Vico-Oton

2013

Int. J. Algebra Comput.

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“…It was proved by Beelen and Ruano in ( [23], Proposition 9) that, if q ∈ Λ, then the Lewittes and the Geil-Matsumoto bounds coincide. For two-generated semigroups, Equation (3) implies that both bounds coincide if and only if q q a b.…”

Section: Coincidences Of Lewittes's and The Geil-matsumoto Boundmentioning

confidence: 99%

Ideals of Numerical Semigroups and Error-Correcting Codes

Bras-Amorós

2019

Symmetry

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Several results relating additive ideals of numerical semigroups and algebraic-geometrycodes are presented. In particular, we deal with the set of non-redundant parity-checks, the codelength, the generalized Hamming weights, and the isometry-dual sequences of algebraic-geometrycodes from the perspective of the related Weierstrass semigroups. These results are related tocryptographic problems such as the wire-tap channel, t-resilient functions, list-decoding, networkcoding, and ramp secret sharing schemes.

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“…This problem, which involves determining the dimension of certain divisors, is challenging and important in its own right. It can be related to many other problems, such as bounding the number of rational points on curves over finite fields [3], [14]. Its study can be further motivated by the construction of algebraic geometry codes, also known as Goppa codes, with good parameters.…”

Section: Introductionmentioning

confidence: 99%

Weierstrass pure gaps on curves with three distinguished points

Borges¹,

Duran²

2021

Preprint

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Let K be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree n + 1 > 3 over K, containing three points, P1, P2, and P3, such that nP1 + P2, nP2 + P3, and nP3 + P1 are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on {P1, P2, P3}, as well as an explicit description of all pure gaps at any subset of {P1, P2, P3}. When K = Fq, this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound.

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Bounding the number of points on a curve using a generalization of Weierstrass semigroups

Abstract: Abstract. In [5] an upper bound for the number of points on an algebraic curve defined over a finite field was derived. In this article we generalize their result by considering Weierstrass groups of several points simultaneously.

Cited by 9 publications

References 10 publications

Nonhomogeneous Patterns on Numerical Semigroups

Nonhomogeneous Patterns on Numerical Semigroups

Ideals of Numerical Semigroups and Error-Correcting Codes

Weierstrass pure gaps on curves with three distinguished points

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