Explicit constructions of Vandermonde sequences using global function fields

Abstract

The authors recently introduced so-called Vandermonde nets. These digital nets share properties with the well-known polynomial lattices. For example, both can be constructed via component-by-component search algorithms. A striking characteristic of the Vandermonde nets is that for fixed $m$ an explicit construction of $m \times m$ generating matrices over the finite field $F_q$ is known for dimensions $s \le q+1$. This paper extends this explicit construction in two directions. We give a maximal extension in terms of $m$ by introducing a construction algorithm for $\infty \times \infty$ generating matrices for digital sequences over $F_q$, which works in the rational function field over $F_q$. Furthermore, we generalize this method to global function fields of positive genus, which leads to extensions in the dimension $s$.