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 \(2 \le q < \infty\).