On the lower bound of the discrepancy of Halton's sequence II

Let $(H_s(n))_{n \geq 1}$ be an $s-$dimensional generalized Halton's sequence. Let $\emph{D}^{*}_N$ be the discrepancy of the sequence $(H_s(n) )_{n = 1}^{N}$. It is known that $D^{*}_{N} =O(\ln^s N)$ as $N \to \infty$. In this paper, we prove that this estimate is exact. Namely, there exists a constant $C(H_s)>0$, such that $$\max_{1 \leq M \leq N} M \emph{D}^{*}_{M} \geq C(H_s) \log_2^s N \quad {\rm for} \; \; N=2,3,... \; .$$