An upper bound for the Garcia-Stichtenoth numbers of towers
type: Item
instance of: scholarly article
title: An upper bound for the Garcia-Stichtenoth numbers of towers
zbMATH Open document ID: 1142.11374
author: Takehiro Hasegawa
published in: Tokyo Journal of Mathematics
publication date: 2006-03-29T00:00:00Z
review text: Let \(F = F/\mathbb{F}_q\) be an algebraic function field in one variable over finite field \(\mathbb{F}_q\), briefly function field. Let \(g(F)\) be the genus of \(F\), and \(N(F)\) of \(\mathbb{F}_q\)-rational places. The author studies the towers \(\mathcal{F} = (F_0 , F_1, F_2, \ldots, F_m, \ldots)\) of function fields where \(F_{m+1}/F_m\) is separable extension of positive degree, and for some \(m \geq 0\), \(F_m/\mathbb{F}_q\) is non-rational and non-elliptic. \textit{A. Garcia} and \textit{H. Stichtenoth} [J. Number Theory 61, No. 2, 248--273 (1996; Zbl 0893.11047)] proved that the limit \(\lambda(\mathcal{F}) = \lambda(\mathcal{F}/\mathbb{F}_q) = \lim_{m \to \infty} N(F_m)/g(F_m)\) exists; it is called Garcia-Stichtenoth number. The author obtains the upper bound for Garcia-Stichtenoth numbers, computes this number for several towers of function fields, and examines several examples, among them the examples of optimal towers.
reviewed by: Vasyl I. Andriychuk
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94B40
zbMATH DE Number: 5015941
zbMATH Keywords: algebraic function fields over finite fields
zbMATH Keywords: Kummer extensions
zbMATH Keywords: coding theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208202
OpenAlex ID: W2070880361
cites work: A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound
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cites work: Q4485863
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DOI: 10.3836/TJM/1244208202
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description: scientific article; zbMATH DE number 5015941