# zbMATH — the first resource for mathematics

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
##### Keywords:
algebraic curves over finite fields
Full Text:
##### References:
 [1] [Russian Text Ignored] 1982. [2] Deligne, Lect. Notes in Math. N 349 pp 143– (1973) · doi:10.1007/978-3-540-37855-6_4 [3] Travaux de Shimura Sém. Bourbaki, Année 1970/71, n$$\deg$$ 389. [4] Ihara, On congruence monodromy problems 2 (1968) [5] On modular curves over finite fields. in ”Discrete subgroups of Lie groups and applications to moduli”, Oxford 1975. [6] Ihara, Proc. Symp. Pure Math. 33 pp 291– (1979) · doi:10.1090/pspum/033.2/546621 [7] [Russian Text Ignored] 1978. [8] [Russian Text Ignored] 41 pp 1008– (1977) [9] Elliptic Functions. Reading. 1973. [10] Langlands, Alg. de Rennes 1978, astérisque 65 pp 125– [11] Über die schlechte Reduktion einiger Shimuramannigfaltigkeiten, Comp. Math. 1981.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.