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

Let $(\bx(n))_{n \geq 1}$ be an $s-$dimensional Niederreiter-Xing sequence in base $b$. Let $D((\bx(n))_{n = 1}^{N})$ be the discrepancy of the sequence $(\bx(n))_{n = 1}^{N}$. It is known that $N D((\bx(n))_{n = 1}^{N}) =O(\ln^s N)$ as $N \to \infty$. In this paper, we prove that this estimate is exact. Namely, there exists a constant $K>0$, such that $$\inf_{\bw \in [0,1)^s} \sup_{1 \leq N \leq b^m} N D((\bx(n)\oplus \bw)_{n = 1}^{N}) \geq K m^s \quad {\rm for} \; \; m=1,2,...\;.$$ We also get similar results for other explicit constructions of $(t,s)$ sequences.