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Let $(H_s(n))_{n \geq 1}$ be an $s-$dimensional Halton's sequence, and let${\mathcal{H}}_{s+1,N}=(H_s(n),n/N)_{n=0}^{N-1}$ be the $s+1-$dimensionalHammersley point set. Let $D(\mathbf{x},(H_n)_{n=0}^{N-1} )$ be the localdiscrepancy of $(H_n)_{n=0}^{N-1}$, and let $D_{s,p} ( (H_n)_{n=0}^{N-1}) $ bethe $L_p$ discrepancy of $(H_n)_{n=0}^{N-1} $. It is known that $\limsup_{N \to\infty} N (\log N)^{-s/2} D_{s,p} (H_s(N))_{n=0}^{N-1} >0$. In this paper, weprove that $$D_{s,p} ((H_s(N))_{n=0}^{N-1}) = O(N^{-1} \log^{s/2} N) \quad {\rmfor} \; \; N \to \infty.$$ I.e., we found the smallest possible order ofmagnitude of $L_p$ discrepancy of Halton's sequence. Then we prove the CentralLimit Theorem for Hammersley net : \begin{equation}\nonumber N^{-1}D(\bar{\mathbf{x}},\mathcal{H}_{s+1,N} )/ D_{s+1,2}(\mathcal{H}_{s+1,N})\stackrel{w}{\rightarrow} \mathcal{N}(0,1), \end{equation} where$\bar{\mathbf{x}}$ is a uniformly distributed random variable in $[0,1]^{s+1}$.The main tool is the theorem on $p$-adic logarithmic forms.

On the upper bound of the $L_p$ discrepancy of Halton's sequence and the Central Limit Theorem for Hammersley's net