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Lattice rules are among the most prominently studied quasi-Monte Carlomethods to approximate multivariate integrals.  A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fullyspecified by its \emph{generating vector} $\boldsymbol{z} \in \mathbb{Z}^s$ andits number of points~$N$.  While there are many results on the existence of ``good'' rank-$1$ latticerules, there are no explicit constructions of good generating vectors fordimensions $s \ge 3$.  This is why one usually resorts to computer search algorithms. In a recentpaper by Ebert et al. in the Journal of Complexity, we showed acomponent-by-component digit-by-digit (CBC-DBD)  construction for good generating vectors of rank-1 lattice rules forintegration of functions in weighted Korobov classes. However, the result inthat paper was  limited to product weights. In the present paper, we shall generalize thisresult to arbitrary positive weights, thereby answering an open question posed  in the paper of Ebert et al. We also include a short section on how thealgorithm can be implemented in the case of POD weights, by which we see thatthe CBC-DBD construction is competitive with the classical CBC construction.

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