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Good low degree rank-1 lattice rules of high dimension

Author: Tor Sørevik
Journal: Math. Comp. 85 (2016), 1821-1835
MSC (2010): Primary 65D32; Secondary 42A10
Published electronically: January 7, 2016
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Abstract: In this paper we introduce a novel approach to searching for rank-1 lattice rules. The idea is to separate the search into two steps, first finding good generating vectors and then finding the corresponding optimal $ N$ value. For the trigonometric degree $ \delta = 5$ we establish a simple criterion on the generating vectors. By using the theory for Golomb rulers and $ {\mathcal B}_2$-series we construct efficient algorithms for finding good generating vectors. Combined with our own home-brewed algorithm for finding the corresponding optimal $ N$, we produce new good rank-1 lattice rules of high dimension.

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Tor Sørevik
Affiliation: Department of Mathematics, University of Bergen, Bergen, Norway

Keywords: Optimal lattice rules, trigonometric degree, Golomb rulers
Received by editor(s): February 27, 2014
Received by editor(s) in revised form: October 29, 2014, and January 7, 2015
Published electronically: January 7, 2016
Dedicated: In memory of James N. Lyness
Article copyright: © Copyright 2016 American Mathematical Society