Designs, Codes and Cryptography

, Volume 70, Issue 1–2, pp 117–125

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

Preview

Unable to display preview. Download preview PDF.

References

1. 1.
Beelen P., Ruano D.: Bounding the number of points on a curve using a generalization of Weierstrass semigroup. Des. Codes Cryptogr. (2012) (Accepted).Google Scholar
2. 2.
García-Sánchez P.A., Rosales J.C.: Numerical semigroups generated by intervals. Pac. J. Math. 191(1), 75–83 (1999)
3. 3.
Geil O.: On codes from norm-trace curves. Finite Fields Appl. 9(3), 351–371 (2003)
4. 4.
Geil O., Matsumoto R.: Bounding the number of $${\mathbb{F}_q}$$ -rational places in algebraic function fields using Weierstrass semigroups. J. Pure Appl. Algebra 213(6), 1152–1156 (2009)
5. 5.
Høholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Handbook of Coding Theory, vol. I, II, pp. 871–961. North-Holland, Amsterdam (1998).Google Scholar
6. 6.
Kirfel C., Pellikaan R.: The minimum distance of codes in an array coming from telescopic semigroups. IEEE Trans. Inform. Theory 41(6 part 1), 1720–1732 (1995) (Special issue on algebraic geometry codes).Google Scholar
7. 7.
Lewittes J.: Places of degree one in function fields over finite fields. J. Pure Appl. Algebra 69(2), 177–183 (1990)
8. 8.
Rosales J.C., García-Sánchez P.A.: Numerical semigroups. In: Developments in Mathematics, vol. 20. Springer, New York (2009).Google Scholar
9. 9.
Stichtenoth H.: Algebraic function fields and codes. Universitext, Springer-Verlag, Berlin (1993)