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