Coding Theory and Design Theory pp 137-162 | Cite as

# Algebraic Geometric Codes

## 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 F_{2} will be possible; (this author is not one of them).

## Keywords

Rational Point Linear Code Algebraic Curf Cyclic Code Projective Curve## Preview

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## References

- [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]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]Y.Driencourt and J.F.Michon,
*Rapport sur les Codes Géométriques*, Univ.Aix-Marseille II et Université Paris 7.Google Scholar - [4]
- [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]V.D.Goppa,
*A new class of linear error-correcting codes*, Probl.of Information Transmission, 6 (1970), pp. 207–212.MathSciNetGoogle Scholar - [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]
- [9]
- [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]F.J.Mac Williams and N.J.A.Sloane,
*The Theory of Error-Correcting Codes*, North Holland, Amsterdam, 1977.Google Scholar - [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]A.N.Skorobogatov and S.G.Vlädut,
*On the Decoding of Algebraic-Geometric Codes*, preprint.Google Scholar - [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]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]M.Wirtz,
*Verallgemeinerte Goppa-Codes*, Diplomarbeit University of Munster (W.Germany).Google Scholar