Some properties of rank-$2$ lattice rules

A rank-2 lattice rule is a quadrature rule for the (unit) s-dimensional hypercube, of the form \[ Qf = (1/{n_1}{n_2})\sum \limits _{{j_1} = 1}^{{n_1}} {\sum \limits _{{j_2} = 1}^{{n_2}} {\bar f({j_1}{{\mathbf {z}}_1}/{n_1} + {j_2}{{\mathbf {z}}_2}/{n_2}),} } \] which cannot be re-expressed in an analogous form with a single sum. Here $\bar f$ is a periodic extension of f, and ${{\mathbf {z}}_1}$, ${{\mathbf {z}}_2}$ are integer vectors. In this paper we discuss these rules in detail; in particular, we categorize a special subclass, whose leading one- and two-dimensional projections contain the maximum feasible number of abscissas. We show that rules of this subclass can be expressed uniquely in a simple tricycle form.