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Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes

Authors: A. Campillo and J. I. Farrán
Journal: Math. Comp. 71 (2002), 1759-1780
MSC (2000): Primary 14Q05; Secondary 11T71
Published electronically: December 4, 2001
MathSciNet review: 1933054
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Abstract: In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at $P$ together with functions for each value in this semigroup, provided $P$ is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes.

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  • 1. Antonio Campillo, Algebroid curves in positive characteristic, Lecture Notes in Mathematics, vol. 813, Springer, Berlin, 1980. MR 584440
  • 2. A. Campillo and J. I. Farrán, Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models, Finite Fields Appl. 6 (2000), no. 1, 71–92. MR 1738217,
  • 3. E. Casas-Alvero, Infinitely near imposed singularities and singularities of polar curves, Math. Ann. 287 (1990), no. 3, 429–454. MR 1060685,
  • 4. E. Casas-Alvero, ``Singularities of plane curves", London Math. Soc. Lecture Notes Series 276, Cambridge University Press (2000). CMP 2001:01
  • 5. Dominique Duval, Rational Puiseux expansions, Compositio Math. 70 (1989), no. 2, 119–154. MR 996324
  • 6. F. Enriques and O. Chisini, ``Teoria geometrica delle equazioni e delle funzioni algebriche", Bologna (1918).
  • 7. J.I. Farrán and Ch. Lossen, ``brnoeth.lib", A SINGULAR 2.0 library for the Brill-Noether algorithm, Weierstrass semigroups and AG codes (2001). Available via http://www.singular.
  • 8. Gui Liang Feng and T. R. N. Rao, Decoding algebraic-geometric codes up to the designed minimum distance, IEEE Trans. Inform. Theory 39 (1993), no. 1, 37–45. MR 1211489,
  • 9. V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259 (1981), no. 6, 1289–1290 (Russian). MR 628795
  • 10. V. D. Goppa, Algebraic-geometric codes, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 4, 762–781, 896 (Russian). MR 670165
  • 11. D. Gorenstein, ``An arithmetic theory of adjoint plane curves", Trans. Amer. Math. Soc. 72, pp. 414-436 (1952). MR 14:198h
  • 12. G.-M. Greuel, G. Pfister and H. Schönemann, `` SINGULAR 2.0", A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2001). Available via
  • 13. G. Haché, ``Construction effective des codes géométriques", Ph.D. thesis, Univ. Paris 6 (1996).
  • 14. Gaétan Haché, Computation in algebraic function fields for effective construction of algebraic-geometric codes, Applied algebra, algebraic algorithms and error-correcting codes (Paris, 1995) Lecture Notes in Comput. Sci., vol. 948, Springer, Berlin, 1995, pp. 262–278. MR 1448169,
  • 15. Gaétan Haché and Dominique Le Brigand, Effective construction of algebraic geometry codes, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1615–1628. Special issue on algebraic geometry codes. MR 1391019,
  • 16. Johan P. Hansen and Henning Stichtenoth, Group codes on certain algebraic curves with many rational points, Appl. Algebra Engrg. Comm. Comput. 1 (1990), no. 1, 67–77. MR 1325513,
  • 17. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • 18. Tom Høholdt and Ruud Pellikaan, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1589–1614. Special issue on algebraic geometry codes. MR 1391018,
  • 19. M.D. Huang and D. Ierardi, ``Efficient algorithms for Riemann-Roch problem and for addition in the jacobian of a curve", Proceedings 32nd Annual Symposium on Foundations of Computer Sciences, pp. 678-687, IEEE Comput. Soc. Press (1991).
  • 20. Christoph Kirfel and Ruud Pellikaan, The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1720–1732. Special issue on algebraic geometry codes. MR 1391031,
  • 21. D. Le Brigand and J.-J. Risler, Algorithme de Brill-Noether et codes de Goppa, Bull. Soc. Math. France 116 (1988), no. 2, 231–253 (French, with English summary). MR 971561
  • 22. Joseph Lipman, On complete ideals in regular local rings, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 203–231. MR 977761
  • 23. Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • 24. M. Rybowicz, ``Sur le calcul des places et des anneaux d'entiers d'un corps de fonctions algébriques", Ph.D. thesis, Limoges (1990).
  • 25. M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric codes, Mathematics and its Applications (Soviet Series), vol. 58, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian by the authors. MR 1186841

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Additional Information

A. Campillo
Affiliation: Departamento de Algebra, Geometría y Topología, Universidad de Valladolid, Spain

J. I. Farrán
Affiliation: Departamento de Matemática Aplicada a la Ingeniería, Universidad de Valladolid, Spain

Keywords: Algebraic curves, singular plane models, desingularization, symbolic Hamburger-Noether expressions, adjoints, virtual passing conditions, Weierstrass semigroups, AG codes
Received by editor(s): October 13, 1999
Received by editor(s) in revised form: December 26, 2000
Published electronically: December 4, 2001
Additional Notes: Both authors are partially supported by DIGICYT PB97-0471.
Article copyright: © Copyright 2001 American Mathematical Society