  Given a finite set $A \subseteq \mathbb{R}^d$, points
$a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the
vertices of a convex polytope which contains no points of $A$ in its interior.
We construct arbitrarily large point sets in general position in $\mathbb{R}^d$
having no holes of size $O(4^dd\log d)$ or more. This improves the previously
known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our
construction uses a certain type of equidistributed point sets, originating
from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding
a bound of $2^{7d}$. The better bound is obtained using a variant of
$(t,m,s)$-nets, obeying a relaxed equidistribution condition.
