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Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound. (English) Zbl 0574.94013
A linear code is a \(k\)-dimensional subspace \(C\subset\mathbb F^ n_ q\). Let \(d\) be the minimum number of nonzero coordinates for nonzero elements of \(C\). One of the main problems of coding theory is to construct infinite series of codes over fixed \(\mathbb F_ q\) with \(\delta =d/n\) and \(R=k/n\) as large as possible. More than twenty years ago so called Varshamov-Gilbert codes were considered best possible in that sense. In 1982 V. D. Goppa discovered a broad class of codes arising from algebraic curves over finite fields. In this work the existence of Goppa codes better than Varshamov-Gilbert ones in some interval of \(\delta\) is established.
Reviewer: A.Givental’

MSC:
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
14G50 Applications to coding theory and cryptography of arithmetic geometry
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References:
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