In many applications the dimension s can be rather large. But, in this case, the asymptotically almost optimal bounds on the discrepancy which we obtained, e.g., for the Hammersley point set or for (t, m, s)-nets soon become useless for a modest number N of points. For example, assume that for every \(s,N \in {\mathbb N}\) we have a point set \(\mathcal {P}_{s,N}\) in the s-dimensional unit cube of cardinality N with star discrepancy of at most \(\begin{aligned} D_N^{*}(\mathcal {P}_{s,N}) \le c_s \frac{(\log N)^{s-1}}{N} \end{aligned}\) with some quantity \(c_s>0\) that is independent of N. Hence for any \(\delta >0\) the star discrepancy behaves asymptotically like \(N^{-1+\delta }\) , which is the optimal rate of convergence since for dimension \(s = 1\) we already have \(D^*_N(\mathcal {P}_{1,N}) \ge 1/(2N)\) . However, the function \(N \mapsto (\log N)^{s-1}/N\) does not start to decrease to zero until \(N \ge \exp (s-1)\) . For \(N \le \exp (s-1)\) this function is increasing, which means that for cardinality N in this range our discrepancy bounds are useless.