On the existence of higher order polynomial lattices based on a generalized figure of merit

Dick and Pillichshammer recently introduced generalized rank-1 polynomial lattices which can be viewed as digital (t,α,β,n×m,s)-nets as introduced by the first author. In this work we generalize the figure of merit of rank-1 polynomial lattices such that the new figure of merit ϱα is related to the t-value, when one views the rank-1 polynomial lattice as a digital (t,α,β,n×m,s)-net. Then we show the existence of rank-1 polynomial lattices for which the generalized figure of merit ϱα satisfies a certain condition. We present some numerical results comparing the corresponding t-value to known explicit constructions.