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 $2\le q<\infty$.

References in corpus (1)

• Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order

Cited by in corpus (1)

• On the $L_p$ discrepancy of two-dimensional folded Hammersley point sets