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Designs, Codes and Cryptography

, Volume 66, Issue 1–3, pp 221–230 | Cite as

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

  • Peter Beelen
  • Diego Ruano
Article

Abstract

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 (generalized) Weierstrass semigroup (J Pure Appl Algebra 207(2), 243–260, 2006) for an n-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in (J Pure Appl Algebra 213(6), 1152–1156, 2009).

Keywords

Algebraic function field Rational place Linear code AG-code Weierstrass semigroup 

Mathematics Subject Classification

14G15 11G20 14H05 14H25 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.DTU-MathematicsTechnical University of DenmarkLyngbyDenmark
  2. 2.Department of Mathematical SciencesAalborg UniversityAalborg ØstDenmark

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