Lattice rules of minimal and maximal rank with good figures of merit

For periodic integrands with unit period in each variable, certain error bounds for lattice rules are conveniently characterised by the figure of merit ρ, which was originally introduced in the context of number theoretic rules. The problem of finding good rules of order N (that is, having N distinct nodes) then becomes that of finding rules with large values of ρ. This paper presents efficient search methods for the discovery of rank 1 rules, and of maximal rank rules of high order, which possess good figures of merit.