We give a new description of the so-called hyperbolic codes from which the minimum distance and the generator matrix are easily determined. We also give a method for the determination of the dimension of the codes and finally some results on the weight hierarchy are presented.


Hyperbolic codes generalized Hamming weights 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Cox, J. Little, and D. O’shea, Ideals, Varieties, and Algorithms, 2nd ed., Springer, Berlin 1997.Google Scholar
  2. 2.
    G.-L. Feng and T.R.N. Rao, “Improved Geometric Goppa Codes, Part I:Basic theory,” IEEE Trans. Inform. Theory, vol. 41, pp. 1678–1693, Nov. 1995.Google Scholar
  3. 3.
    G.-L. Feng, T. R. N. Rao, G. A. Berg, and J. Zhu, “Generalized Bezout’s Theorem and Its Applications in Coding Theory”, IEEE Trans. Inform. Theory, vol. 43, pp. 1799–1810, Nov. 1997.Google Scholar
  4. 4.
    G.-L. Feng, J. Zhu, X. Shi, and T. R. N. Rao, “The Applications of Generalized Bezout’s Theorem to the Codes from the Curves in High Dimensional Spaces”, in Proc. 35th Allerton Conf. Communication, Control and Computing, pp. 205–214, 1997.Google Scholar
  5. 5.
    O. Geil, “On the Construction of Codes from Order Domains”, submitted to IEEE Trans. Inform. Theory, June. 2001.Google Scholar
  6. 6.
    O. Geil, and T. Høholdt, “Footprints or Generalized Bezout’s Theorem”, IEEE Trans. Inform. Theory, vol. 46, pp. 635–641, Mar. 2000.Google Scholar
  7. 7.
    P. Heijnen, and R. Pellikaan, “Generalized Hamming weights of q-ary Reed-Muller codes”, IEEE Trans. Inform. Theory, vol. 44, pp. 181–196, Jan. 1998.Google Scholar
  8. 8.
    T. Høholdt, J. H. van Lint, and R. Pellikaan, “Algebraic Geometry Codes”, in Handbookof Coding Theory, (V. S. Pless, and W. C. Hufman Eds.), vol 1, pp. 871–961, Elsevier, Amsterdam 1998.Google Scholar
  9. 9.
    K. Saints, and C. Heegard, “On Hyperbolic Cascaded Reed-Solomon codes”, Proc. AAECC-10, Lecture Notes in Comput. Sci. Vol. 673, pp. 291–303, Springer, Berlin 1993.Google Scholar
  10. 10.
    T. Shibuya and K. Sakaniwa, “A Dual of Well-Behaving Type Designed Minimum Distance,” IEICE Trans. A, vol. E84-A, no. 2, pp. 647–652, Feb. 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Olav Geil
    • 1
  • Tom Høholdt
    • 2
  1. 1.Department of Mathematical SciencesAalborg UniversityAalborg ØDenmark
  2. 2.Department of MathematicsTechnical University of DenmarkLyngbyDenmark

Personalised recommendations