  The star discrepancy $D_N^*(\mathcal{P})$ is a quantitative measure for the
irregularity of distribution of a finite point set $\mathcal{P}$ in the
multi-dimensional unit cube which is intimately related to the integration
error of quasi-Monte Carlo algorithms. It is known that for every integer $N
\ge 2$ there are point sets $\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$
and $D_N^*(\mathcal{P}) =O((\log N)^{d-1}/N)$. However, for small $N$ compared
to the dimension $d$ this asymptotically excellent bound is useless (e.g. for
$N \le {\rm e}^{d-1}$).
  In 2001 it has been shown by Heinrich, Novak, Wasilkowski and
Wo\'{z}niakowski that for every integer $N \ge 2$ there exist point sets
$\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$ and $D_N^*(\mathcal{P}) \le C
\sqrt{d/N}$. Although not optimal in an asymptotic sense in $N$, this upper
bound has a much better (and even optimal) dependence on the dimension $d$.
  Unfortunately the result by Heinrich et al. and also later variants thereof
by other authors are pure existence results and until now no explicit
construction of point sets with the above properties are known. Quite recently
L\"obbe studied lacunary subsequences of Kronecker's $(n
\boldsymbol{\alpha})$-sequence and showed a metrical discrepancy bound of the
form $C \sqrt{d (\log d)/N}$ with implied absolute constant $C>0$ independent
of $N$ and $d$.
  In this paper we show a corresponding result for digital Kronecker sequences,
which are a non-archimedean analog of classical Kronecker sequences.
