A lower bound on a quantity related to the quality of polynomial lattices

Abstract

In this paper, we study a quantity $R_b$ which is closely related to the quality of an important subclass of digital $(t,m,s)$-nets over a finite field $\mathbb{F}_b$, namely polynomial lattices. Niederreiter has shown by an averaging argument that there always exist generators of polynomial lattices for which $R_b$ is small, establishing thereby the existence of polynomial lattices with particularly low star discrepancy. In this work, we show that this result is best possible, i.e., we prove that for all generators of polynomial lattices the quantity $R_b$ cannot go below a certain threshold.