Optimal L_p-discrepancy bounds for second order digital sequences

The Lp-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 Lp-discrepancy of general infinite sequences in the ddimensional 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 L2-discrepancy matches the lower bound of Proinov. Here we give a complete solution to this problem for all finite p > 1. We consider socalled order 2 digital (t, d)-sequences over the finite field with two elements and show that such sequences achieve the optimal order of Lp-discrepancy simultaneously for all p ∈ (1,∞).