  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
  $$
  \max_{1 \leq M \leq N}
  M \emph{D}^{*}_{M} \geq C(H_s) \log_2^s N \quad {\rm for} \; \; N=2,3,... \; .
  $$
