Designs, Codes and Cryptography

, Volume 70, Issue 1–2, pp 117–125 | Cite as

On the Geil–Matsumoto bound and the length of AG codes

Article

Abstract

The Geil–Matsumoto bound conditions the number of rational places of a function field in terms of the Weierstrass semigroup of any of the places. Lewittes’ bound preceded the Geil–Matsumoto bound and it only considers the smallest generator of the numerical semigroup. It can be derived from the Geil–Matsumoto bound and so it is weaker. However, for general semigroups the Geil–Matsumoto bound does not have a closed formula and it may be hard to compute, while Lewittes’ bound is very simple. We give a closed formula for the Geil–Matsumoto bound for the case when the Weierstrass semigroup has two generators. We first find a solution to the membership problem for semigroups generated by two integers and then apply it to find the above formula. We also study the semigroups for which Lewittes’s bound and the Geil–Matsumoto bound coincide. We finally investigate on some simplifications for the computation of the Geil–Matsumoto bound.

Keywords

Algebraic function field Weierstrass semigroup Geil–Matsumoto bound Gonality bound Lewittes’ bound 

Mathematics Subject Classification

14Q05 11G20 68R01 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Universitat Rovira i VirgiliTarragona, CataloniaSpain

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