Summary
Rank-1 lattice rules based on a weighted star discrepancy with weights of a product form have been previously constructed under the assumption that the number of points is prime. Here, we extend these results to the non-prime case. We show that if the weights are summable, there exist lattice rules whose weighted star discrepancy is O(n −1+δ), for any δ > 0, with the implied constant independent of the dimension and the number of lattice points, but dependent on δ and the weights. Then we show that the generating vector of such a rule can be constructed using a component-by-component (CBC) technique. The cost of the CBC construction is analysed in the final part of the paper.
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Sinescu, V., Joe, S. (2008). Good Lattice Rules with a Composite Number of Points Based on the Product Weighted Star Discrepancy. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_39
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DOI: https://doi.org/10.1007/978-3-540-74496-2_39
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