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Designs, Codes and Cryptography

, Volume 45, Issue 3, pp 359–364 | Cite as

Codes from curves with total inflection points

  • Cícero Carvalho
  • Takao Kato
Article

Abstract

The concept of pure gaps of a Weierstrass semigroup at several points of an algebraic curve has been used lately to obtain codes that have a lower bound for the minimum distance which is greater than the Goppa bound. In this work, we show that the existence of total inflection points on a smooth plane curve determines the existence of pure gaps in certain Weierstrass semigroups. We then apply our results to the Hermitian curve and construct codes supported on several points that compare better to one-point codes from that same curve.

Keywords

Geometric Goppa codes Weierstrass semigroup Pure gap Total inflection point Plane curve Hermitian curve 

AMS Classifications

94B27 14H50 11T71 11G20 

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References

  1. 1.
    Carvalho C., Torres F. (2005). On Goppa codes and Weierstrass gaps at several points. Des. Codes Cryptogr. 35, 211–225MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Coppens M., Kato T. (1994). Weierstrass gap sequences at total inflection points of nodal plane curves. Tsukuba J. Math. 1, 119–129MathSciNetGoogle Scholar
  3. 3.
    Goppa V.D. (1983). Algebraic-geometric codes. Math. USSR-Izv. 21, 75–93CrossRefGoogle Scholar
  4. 4.
    Homma M., Kim S.J. (2001). Goppa codes with Weierstrass pairs. J. Pure Appl. Algebra 162: 273–290MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Pellikaan R.: On special divisors and the two variable zeta function of algebraic curves over finite fields. Proceedings AGCT-4, Luminy, 175–184 (1997).Google Scholar
  6. 6.
    Stichtenoth H. (2001). Algebraic Function Fields and Codes. Springer Universitext, New YorkGoogle Scholar
  7. 7.
    Stichtenoth H. (1988). A note on Hermitian codes over GF(q 2). IEEE Trans. Inform. Theory 34, 1345–1348CrossRefMathSciNetGoogle Scholar
  8. 8.
    Yang K., Kumar P.V. (1992). On the true minimum distance of Hermitian codes. Lect. Notes Math. 1518, 99–107MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal de UberlândiaUberlândiaBrazil
  2. 2.Department of Mathematical Sciences, Faculty of SciencesYamaguchi UniversityYamaguchiJapan

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