An upper bound on the minimal dispersion

For $\epsilon \in (0,1\mathrm{/}2)$ and a natural number $d\ge 2$, let $N$ be a natural number with $$N\ge 2^9{\log }_2(d){\left(\frac{{\log }_2(1\mathrm{/}\epsilon )}{\epsilon }\right)}^2\mathrm{.}$$ We prove that there is a set of $N$ points in the unit cube $[0,1{]}^d$, which intersects all axis-parallel boxes with volume $\epsilon$. That is, the dispersion of this point set is bounded from above by $\epsilon$.