A note on the CBC-DBD construction of lattice rules with general positive weights

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its generating vector $\boldsymbol{z} \in \mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of "good" rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s \ge 3$. This is why one usually resorts to computer search algorithms. In the paper [5], we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in [5].