Nets, \((t,s)\)-sequences, and algebraic curves over finite fields with many rational points
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.
scientific article; zbMATH DE number 1184376