  The $L_p$-discrepancy is a quantitative measure for the irregularity of
distribution modulo one of infinite sequences. In 1986 Proinov proved for all
$p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences in
the $d$-dimensional unit cube, but it remained an open question whether this
lower bound is best possible in the order of magnitude until recently. In 2014
Dick and Pillichshammer gave a first construction of an infinite sequence whose
order of $L_2$-discrepancy matches the lower bound of Proinov. Here we give a
complete solution to this problem for all finite $p > 1$. We consider so-called
order $2$ digital $(t,d)$-sequences over the finite field with two elements and
show that such sequences achieve the optimal order of $L_p$-discrepancy
simultaneously for all $p \in (1,\infty)$.
