Advertisement

Algebraic Geometric Codes

  • Jacobus H. Van Lint
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 20)

Abstract

The most important development in the theory of error-correcting codes in recent years is the introduction of methods from algebraic geometry to construct good codes. The ideas are based on generalizations of so-called Goppa codes. The (by now) “classical” Goppa codes (1970, cf.[6]) were already a great improvement on codes known at that time. The algebraic geometric codes were also inspired by ideas of Goppa but the most sensational development was a paper by Tsfasman, Vlădut and Zink (1982,cf.[15]). In this paper the idea of codes from algebraic curves was combined with certain recent deep results from algebraic geometry to produce a sequence of error-correcting codes that led to a new lower bound on the information rate of good codes that is better than the Gilbert-Varshamov bound. The novice reader should realize that the G-V-bound (1952) was never improved (until 1982) and believed by many to be best possible. Actually, the improvement is only achieved for alphabets of size at least 49 and several binary coding experts still have hope that no improvement of the G-V-bound for F2 will be possible; (this author is not one of them).

Keywords

Rational Point Linear Code Algebraic Curf Cyclic Code Projective Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A.M.Barg, S.L.Katsman and M.A.Tsfasman, Algebraic Geometric Codes from Curves of Small Genus, Probl.of Information Transmission, 23 (1987), pp. 34–38.MathSciNetzbMATHGoogle Scholar
  2. [2]
    T.Beth, Some aspects of coding theory between probability, algebra, combinatorics and complexity theory, in Combinatorial Theory, Lecture Notes in Mathematics 969, Springer Verlag, New York.Google Scholar
  3. [3]
    Y.Driencourt and J.F.Michon, Rapport sur les Codes Géométriques, Univ.Aix-Marseille II et Université Paris 7.Google Scholar
  4. [4]
    W.Fulton, Algebraic Curves, Benjamin Cummings, Reading, 1969.zbMATHGoogle Scholar
  5. [5]
    G. van der Geer and J.H. van Lint, Introduction to Coding Theory and Algebraic Geometry, DMV Lecture Notes (to appear).Google Scholar
  6. [6]
    V.D.Goppa, A new class of linear error-correcting codes, Probl.of Information Transmission, 6 (1970), pp. 207–212.MathSciNetGoogle Scholar
  7. [7]
    J.Justesen, K.J.Larsen, H. Elbrønd Jensen, A.Havemose and T.Høholdt, Construction and decoding of a class of algebraic geometry codes, MAT.Rep.No.1988-10, Danmarks Tekniske Højskole.Google Scholar
  8. [8]
    G.Lachaud, Les codes géométriques de Goppa, Séminaire Bourbaki, no.641.Google Scholar
  9. [9]
    J.H. van Lint, Introduction to Coding Theory, Springer Verlag, New York, 1982.zbMATHGoogle Scholar
  10. [10]
    J.H. van Lint and T.A.Springer, Generalized Reed-Solomon Codes from Algebraic Geometry, IEEE Trans.on Information Theory, IT-33 (1987), pp. 305–309.CrossRefGoogle Scholar
  11. [11]
    F.J.Mac Williams and N.J.A.Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977.Google Scholar
  12. [12]
    J.-P. Serre, Sur le nombre des points rationnels d’une courbe algébrique sur un corps fìni, C.R. Acad. Sc. Paris, 296 (1983), pp. 397–402.zbMATHGoogle Scholar
  13. [13]
    A.N.Skorobogatov and S.G.Vlädut, On the Decoding of Algebraic-Geometric Codes, preprint.Google Scholar
  14. [14]
    M.A.Tsfasman, Goppa codes that are better than the Varshamov-Gilbert bound, Probl.of Information Transmission, 18 (1982), pp. 163–165.Google Scholar
  15. [15]
    M.A.Tsfasman, S.G.Vlädut and T.Zink, Modular curves, Shimura curves and Goppa codes better than the Varshamov- Gilbert bound, Math.Nachr., 109 (1982), pp. 21–28.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    M.Wirtz, Verallgemeinerte Goppa-Codes, Diplomarbeit University of Munster (W.Germany).Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • Jacobus H. Van Lint
    • 1
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenNetherlands

Personalised recommendations