On the lower bound of the discrepancy of (t,s)-sequences: II

Abstract

Let $(x(n){)}_{n\ge 1}$ be an $s$-dimensional Niederreiter-Xing sequence in base $b$. Let $D((x(n){)}_{n=1}^N)$ be the discrepancy of the sequence $(x(n){)}_{n=1}^N$. It is known that $ND((x(n){)}_{n=1}^N)=O({\ln }^{s}N)$ as $N\to \mathrm{\infty }$. In this paper, we prove that this estimate is exact. Namely, there exists a constant $K>0$, such that
$$\underset{w\in [0,1{]}^s}{inf}\underset{1\le N\le b^m}{sup}ND((x(n)\oplus w{)}_{n=1}^N)\ge K{\ln }^s\text{ for }m=1,2,\dots .$$

We also get similar results for other explicit constructions of $(t,s)$-sequences.