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Les codes géométriques de Goppa. (French) Zbl 0603.94010
Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. No. 641, Astérisque 133/134, 189-207 (1986).
[For the entire collection see Zbl 0577.00004.]
The theory of linear error correcting codes is first restated in the manner of Goppa, using the theory of curves in an algebraic geometry over a finite field. Basic properties of codes are developed, including bounds on the size of codes and the definition of an excellent family of codes as a family that asymptotically lies above the Varshamov-Gilbert bound. Results on algebraic curves are reviewed and used to define classes of codes in algebraic geometries and the properties of codes derived in terms of the geometric parameters. The class of Goppa codes is included. The performance of these classes of codes is considered and the problem of decoding them are examined.
Reviewer: I.F.Blake

MSC:
94B05 Linear codes, general
14H99 Curves in algebraic geometry
94B35 Decoding
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