A construction of (t,s)-sequences with finite-row generating matrices using global function fields

For any prime power q and any dimension s⩾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→∞. 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,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.