Three- and four-dimensional K-optimal lattice rules of moderate trigonometric degree

A systematic search for optimal lattice rules of specified trigonometric degree d over the hypercube [0, 1)s has been undertaken. The search is restricted to a population K(s, δ) of lattice rules Q(Λ). This includes those where the dual lattice Λ⊥ may be generated by s points h for each of which |h| = δ = d+ 1. The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for s = 3, d ≤ 29 and s = 4, d ≤ 23, a list of K-optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.