Good low degree rank-1 lattice rules of high dimension

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.