  For any prime power $q$ and any dimension $s \ge 1$, we present a
construction of $(t,s)$-sequences in base $q$ with finite-row generating
matrices such that, for fixed $q$, the quality parameter $t$ is asymptotically
optimal as a function of $s$ as $s \to \infty$. This is the first construction
of $(t,s)$-sequences that yields finite-row generating matrices and
asymptotically optimal quality parameters at the same time. The construction is
based on global function fields. We put the construction into the framework of
$(u,{\bf e},s)$-sequences that was recently introduced by Tezuka. In this way
we obtain in many cases better discrepancy bounds for the constructed sequences
than by previous methods for bounding the discrepancy.
