On the asymptotic behaviour of some towers of function fields over finite fields
type: Item
instance of: scholarly article
title: On the asymptotic behaviour of some towers of function fields over finite fields
zbMATH Open document ID: 0893.11047
author: Arnaldo Garcia
author: Henning Stichtenoth
published in: Journal of Number Theory
publication date: 1997-04-24T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 94B27
zbMATH DE Number: 980268
zbMATH Keywords: towers of function fields
zbMATH Keywords: asymptotically good towers
zbMATH Keywords: Drinfeld-Vladut bound
zbMATH Keywords: Artin-Schreier extension
zbMATH Keywords: long algebraic-geometric codes with good parameters
reviewed by: Robert F. Lax
MaRDI profile type: Publication
full work available at URL: jnth.1996.0147
OpenAlex ID: W1996118782
DOI: 10.1006/JNTH.1996.0147
description: scientific article; zbMATH DE number 980268