Goppa's algebraic-geometry codes over a finite field GF(q) are based on algebraic curves over GF(q) with many rational points. This leads to a well-known connection between the asymptotic behavior of good algebraic-geometry codes and the asymptotics of the number of rational points on algebraic curves with growing genus. In the seminal paper of Tsfasman, Vladut, and Zink from 1982 this was used to beat the asymptotic form of the Gilbert-Varshamov bound in the case where q is a sufficiently large square.

The talk will report on recent joint work of the speaker and C.P. Xing which has produced significant advances in the asymptotic theory of the number of rational points on algebraic curves. The results allow us to beat the asymptotic Gilbert-Varshamov bound in the case where q is a sufficiently large composite nonsquare. Tools from function field theory such as Drinfeld modules and class field towers are important in our approach.

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