By Henning Stichtenoth, Michael A. Tsfasman

ISBN-10: 0387556516

ISBN-13: 9780387556512

ISBN-10: 3540556516

ISBN-13: 9783540556510

Approximately ten years in the past, V.D. Goppa came upon a shocking connection among the idea of algebraic curves over a finite box and error-correcting codes. the purpose of the assembly "Algebraic Geometry and Coding idea" used to be to offer a survey at the current country of analysis during this box and similar themes. The lawsuits comprise learn papers on numerous points of the concept, between them: Codes built from exact curves and from higher-dimensional types, interpreting of algebraic geometric codes, hint codes, Exponen- tial sums, quickly multiplication in finite fields, Asymptotic variety of issues on algebraic curves, Sphere packings.

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Extra info for Coding Theory and Algebraic Geometry: Proceedings of the International Workshop held in Luminy, France, June 17-21, 1991

Sample text

More generally, it enables us to show that a holomorphic section of a complex vector bundle over a complex manifold is defined everywhere if it is defined on the complement of an analytic subset of codimension 2. This is classically used to show the invariance of the “plurigenera” under birational transformations. We conclude this chapter with a proof of an explicit formula for the local solution of the equation ∂f = g, ∂z where g is a differentiable function defined on an open set of C. This will be used in the following chapter, to prove the local exactness of the Dolbeault complex.

Then for every point z 0 in the interior of D, we have the equality f (z 0 ) = 1 2iπ ∂D f (z) dz. 1 Cauchy’s formula and analyticity Let U be an open set of Cn , and let f : U → C be a C 1 map. For u ∈ U , we have a canonical identification TU,u ∼ = Cn . We can thus generalise the notion of a holomorphic function to higher dimensions. 16 The function f is said to be holomorphic if for every u ∈ U , the differential d f u ∈ Hom(TU,u , C) ∼ = Hom(Cn , C) is C-linear. 7 remains true in higher dimensions.

X is thus covered by open sets U such that there exists a submersion φU : U → V, where V is open in R2(n−k) , satisfying ( E)u = Ker φ∗,u : TU,u,R → TR2(n−k) ,φ(u) , ∀u ∈ U. Next, we show that there exists a complex structure on the image of φ for which φ is holomorphic. For this, we first note that if v = φ(u), TV,v = TU,u / E and as E is stable under the endomorphism I corresponding to the almost complex structure on TU , there is an induced complex structure on TV,v . We then show that this complex structure does not depend on the point in the fibre of φ above v.

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Coding Theory and Algebraic Geometry: Proceedings of the International Workshop held in Luminy, France, June 17-21, 1991 by Henning Stichtenoth, Michael A. Tsfasman

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