By Henning Stichtenoth, Michael A. Tsfasman
Approximately ten years in the past, V.D. Goppa came across a stunning connection among the concept of algebraic curves over a finite box and error-correcting codes. the purpose of the assembly "Algebraic Geometry and Coding idea" was once to provide a survey at the current country of analysis during this box and similar issues. The lawsuits include learn papers on numerous facets of the concept, between them: Codes developed from designated curves and from higher-dimensional kinds, deciphering 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 resources for Coding Theory and Algebraic Geometry: Proceedings of the International Workshop held in Luminy, France, June 17-21, 1991
If the step in t would go beyond tf , then the stepsize is adjusted to land exactly on tf . The next box, labeled “Correct,” computes a Newton correction. ” Otherwise, the results of the correction are checked using safety rules deﬁned by Eqs. B and C, invoking “Call safety error,” if either criteria is violated. If the safety rules are met, the algorithm checks for convergence to within 10−τ , either proceeding to update or considering another correction cycle, as necessary. The remaining key parts, namely “Call convergence error,” “Call safety error,” and “Call step success,” are described below.
6]. 20) into account we have εmov (L; x) = sup m s(mL, x) s(mP, x) = sup = ε(P ; x). m m m 2. 1. Upper bounds and submaximal curves. 9]. For 0-dimensional reduced subschemes we have the following result. 1 (Upper bounds). Let X be a smooth projective variety of dimension n and L a nef line bundle on X. Let x1 , . . , xr be r distinct points on X, then n n L . ε(X, L; x1 , . . , xr ) r In particular for a single point x we always have √ n ε(X, L; x) Ln . Proof. Let f : Y −→ X be the blowup x1 , .
According to Grothendieck [SGA2], Expos´e X, one says that the pair (X, Y ) satisﬁes the Grothendieck–Lefschetz condition Lef(X, Y ) if for every open subset U of X containing Y and for every vector bundle E on U the natural map H 0 (U, E) → ˆ is an isomorphism, where E ˆ = π ∗ (E), with π : X/Y → U the canonical H 0 (X/Y , E) morphism. We also say that (X, Y ) satisﬁes the eﬀective Grothendieck-Lefschetz condition Leﬀ(X, Y ) if Lef(X, Y ) holds and, moreover, for every formal vector bundle E on X/Y there exists an open subset U of X containing Y and a vector bundle ˆ E on U such that E ∼ = E.