The existence of good extensible rank-1 lattices

Authors:
Fred J. Hickernell;Harald Niederreiter
Affiliations:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China;Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore
Venue:
Journal of Complexity
Year:
2003

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Abstract

Extensible integration lattices have the attractive property that the number of points in the node set may be increased while retaining the existing points. It is shown here that there exist generating vectors, h, for extensible rank-1 lattices such that for n = b, b2, ... points and dimensions s = 1, 2, ... the figures of merit Rα, Pα and discrepancy are all small. The upper bounds obtained on these figures of merit for extensible lattices are some power of log n worse than the best upper bounds for lattices where h is allowed to vary with n and s.