On a problem of Garcia, Stichtenoth, and Thomas.
type: Item
instance of: scholarly article
title: On a problem of Garcia, Stichtenoth, and Thomas.
zbMATH Open document ID: 1041.11046
published in: Finite Fields and their Applications
publication date: 2002-08-15T00:00:00Z
review text: A tower \(\cdots\rightarrow C_n\rightarrow C_{n-1} \rightarrow \cdots\rightarrow C_1\rightarrow C_0\) of covers of curves \(C_n/\mathbb F_q\) is asymptotically good if \(\lim \#C_n(\mathbb F_q)/g(C_n)>0\), where \(\#C_n(\mathbb F_q)\) is the number of \(\mathbb F_q\)-rational points of \(C_n\) and \(g(C_n)\) is the genus of \(C_n\). Asymptotically good towers are important in coding theory, but then the curves have to be in explicit form. In this short paper the author answers a question arising from the work of \textit{A. Garcia}, \textit{H. Stichtenoth}, and \textit{M. Thomas} [Finite Fields Appl. 3, 257--274 (1997; Zbl 0946.11029)] on the construction of explicit asymptotically good towers. Garcia et al. obtained a simple explicit example over any non-prime field by applying their general result (Theorem 2.2) which asserts that an asymptotically good tower can be constructed via a polynomial \(f(t)\) with certain properties; they show that such \(f(t)\) exists over any non-prime field, but they cannot find such \(f(t)\) over a prime field. The author shows (Theorem 2) that over a prime field there does not exist any polynomial \(f(t)\) satisfying the conditions of Theorem 2.2 of Garcia et al., so their construction cannot be used to obtain asymptotically good towers over prime fields.
reviewed by: M. Rafiq Omar
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1781824
zbMATH Keywords: finite fields
zbMATH Keywords: polynomials
zbMATH Keywords: curves with many points
author: H. W. jun. Lenstra
MaRDI profile type: Publication
full work available at URL: ffta.2001.0331
OpenAlex ID: W2059715984
cites work: On towers and composita of towers of function fields over finite fields
cites work: Q3324075
DOI: 10.1006/FFTA.2001.0331
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description: scientific article; zbMATH DE number 1781824