Nets, \((t,s)\)-sequences, and algebraic curves over finite fields with many rational points
type: Item
instance of: scholarly article
title: Nets, \((t,s)\)-sequences, and algebraic curves over finite fields with many rational points
zbMATH Open document ID: 0899.11038
author: Harald Niederreiter
published in: Documenta Mathematica
publication date: 1998-08-05T00:00:00Z
full work available at URL: https://eudml.org/doc/232019
full work available at URL: Niederreiter.MAN.html
review text: A \((t,m,s)\)-net in base \(b\) is a set of \(b^m\) points in the \(s\)-dimensional unit cube \(I^s\) that satisfy certain strong uniformity properties with regard to their distribution in \(I^s\). A \((t,s)\)-sequence is a sequence of points in \(I^s\) that can be used to form \((t,m,s)\)-nets for all \(m>t\). The author introduced these concepts in 1987 [\textit{H. Niederreiter}, Monatsh. Math. 104, 273-337 (1987; Zbl 0626.10045)], and here gives a survey of the current status of the theory. After discussing links with classical combinatorial objects, such as mutually orthogonal latin squares, the author proceeds to recent uses of algebraic curves with many rational points over a finite field to construct \((t,s)\)-sequences. These applications of curves (or function fields) to nets and sequences have been carried out by \textit{H. Niederreiter} and \textit{C. Xing} [for example, in Acta Arith. 72, No. 3, 281-298 (1995; Zbl 0833.11035)]. The author also discusses current bounds on the number of rational points on curves over a finite field.
Mathematics Subject Classification ID: 11K45
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 94B65
zbMATH DE Number: 1184376
zbMATH Keywords: \((t,s)\)-sequences
zbMATH Keywords: \((t,m,s)\)-nets
zbMATH Keywords: survey
zbMATH Keywords: orthogonal latin squares
zbMATH Keywords: orthogonal array
zbMATH Keywords: rational points
zbMATH Keywords: algebraic-geometric codes
zbMATH Keywords: bounds
zbMATH Keywords: curves over a finite field
reviewed by: Robert F. Lax
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184376