  In 2001 Heinrich, Novak, Wasilkowski and Wo\'zniakowski proved that the
inverse of the star discrepancy satisfies $n(d,\varepsilon)\leq c_{\abs}d
\varepsilon^{-2}$ by showing that there exists a set of points in $[0,1)^d$
whose star-discrepancy is bounded by $c_{\abs}\sqrt{d/N}$. This result was
generalized by Aistleitner who showed that there exists a double infinite
random matrix with elements in $[0,1)$ which partly are coordinates of elements
of a Halton sequence and partly independent uniformly distributed random
variables such that any $N\times d$-dimensional projection defines a set
$\{x_1,\ldots,x_N\}\subset [0,1)^d$ with \begin{equation*}
  D^*_N(x_1,\ldots,x_N)\leq c_{\abs}\sqrt{d/N}. \end{equation*} In this paper
we consider a similar double infinite matrix where the elements instead of
independent random variables are taken from a certain multivariate lacunary
sequence and prove that with high probability each projection defines a set of
points which has up to some constant the same upper bound on its
star-discrepancy but only needs a significantly lower number of digits to
simulate.
