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

Authors:
Josef Dick;Peter Kritzer;Friedrich Pillichshammer;Wolfgang Ch. Schmid
Affiliations:
University of New South Wales Asia, 1 Kay Siang Road, Singapore 248922, Singapore;Fachbereich Mathematik, Universität Salzburg, Hellbrunnerstraße 34, A-5020 Salzburg, Austria;Institut für Finanzmathematik, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria;Fachbereich Mathematik, Universität Salzburg, Hellbrunnerstraße 34, A-5020 Salzburg, Austria
Venue:
Journal of Complexity
Year:
2007

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Abstract

Dick and Pillichshammer recently introduced generalized rank-1 polynomial lattices which can be viewed as digital (t,@a,@b,nxm,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 @r"@a is related to the t-value, when one views the rank-1 polynomial lattice as a digital (t,@a,@b,nxm,s)-net. Then we show the existence of rank-1 polynomial lattices for which the generalized figure of merit @r"@a satisfies a certain condition. We present some numerical results comparing the corresponding t-value to known explicit constructions.