Discrepancy bounds for infinite-dimensional order two digital sequences over $\mathbb{F}_2$

Abstract

In this paper we provide explicit constructions of digital sequences over the finite field of order 2 in the infinite dimensional unit cube whose first $N$ points projected onto the first $s$ coordinates have $\mathcal{L}_q$ discrepancy bounded by $r^{3/2-1/q} \sqrt{m_1^{s-1} + m_2^{s-1} + \cdots + m_r^{s-1}} N^{-1}$ for all $N = 2^{m_1} + 2^{m_2} + \cdots + 2^{m_r} \ge 2$ and $2 \le q < \infty$. In particular we have for $N = 2^m$ that the $\mathcal{L}_q$ discrepancy is of order $m^{(s-1)/2} 2^{-m}$ for all $2 \le q < \infty$.