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Quasi-Monte Carlo (QMC) is an essential tool for integral approximation,Bayesian inference, and sampling for simulation in science, etc. In the QMCarea, the rank-1 lattice is important due to its simple operation, and niceproperties for point set construction. However, the construction of thegenerating vector of the rank-1 lattice is usually time-consuming because of anexhaustive computer search. To address this issue, we propose a simpleclosed-form rank-1 lattice construction method based on group theory. Ourmethod reduces the number of distinct pairwise distance values to generate amore regular lattice. We theoretically prove a lower and an upper bound of theminimum pairwise distance of any non-degenerate rank-1 lattice. Empirically,our methods can generate a near-optimal rank-1 lattice compared with theKorobov exhaustive search regarding the $l_1$-norm and $l_2$-norm minimumdistance. Moreover, experimental results show that our method achieves superiorapproximation performance on benchmark integration test problems and kernelapproximation problems.

Subgroup-based Rank-1 Lattice Quasi-Monte Carlo