On the asymptotic behaviour of some towers of function fields over finite fields
For an infinite tower of function fields \(F_1\subset F_2\subset\dots\) over the finite field \({\mathbb{F}}_q\), the authors show that the sequence \(\{N(F_n)/g(F_n)\}\), where \(N(F_n)\) denotes the number of places of degree 1 of \(F_n/{\mathbb{F}}_q\) and \(g(F_n)\) denotes the genus of \(F_n\), converges to a limit \(\lambda\) as \(g(F_n)\) approaches infinity. They give sufficient conditions on the tower for \(\lambda\) to be zero and sufficient conditions for \(\lambda\) to be positive, in which case the tower is called asymptotically good. (An asymptotically good tower can be used to define long algebraic-geometric codes with good parameters.) In an earlier work [\textit{A. Garcia} and \textit{H. Stichtenoth}, Invent. Math. 121, 211-222 (1995; Zbl 0822.11078)], the authors gave an explicit example of an asymptotically good tower that achieves the Drinfeld-Vladut bound. Here, they give another, somewhat simpler, example of a tower with this property. In the final section, they show that two interesting towers, one studied by Feng and Rao and one suggested by Serre, each have \(\lambda=0\). Asymptotically good towers of Kummer extensions have been given by the authors in [C. R. Acad. Sci., Paris, Sér. I 322, No. 11, 1067-1070 (1996; Zbl 0867.11042)].
long algebraic-geometric codes with good parameters
scientific article; zbMATH DE number 980268