Cyclic Digital Nets, Hyperplane Nets, and Multivariate Integration in Sobolev Spaces

Abstract

Cyclic nets are a special case of digital nets and were recently introduced by Niederreiter. Here we present a construction algorithm for such nets, where we use the root mean square worst-case error of a randomly digitally shifted point set in a weighted Sobolev space as a selection criterion. This yields a feasible construction algorithm since for a cyclic net with qm points (with fixed bijections and fixed ground field) there are qm possible choices. Our results here match the convergence rate and strong tractability results for polynomial lattice rules, hence providing us with an alternative construction algorithm. Further, we improve upon previous results by including constructions over arbitrary finite fields and an arbitrary choice of bijections.