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

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 $r^{3/2-1/q} \sqrt{m_1^{s-1} + m_2^{s-1} + \cdots + m_r^{s-1}} N^{-1}$forall$N = 2^{m_1} + 2^{m_2} + \cdots + 2^{m_r} \ge 2$and$2 \le q < \infty$\mathrm{.}Inparticularwehavefor$N = 2^m$thatthe$$\mathcal{L}$_q$discrepancyisoforder$m^{(s-1)/2} 2^{-m}$forall$2 \le q < \infty$.