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

A systematic search for optimal lattice rules of specified trigonometric de gree d over the hypercube (0, 1)(s) has been undertaken. The search is rest ricted to a population K(s, delta) of lattice rules Q(A). This includes tho se where the dual lattice Lambda (perpendicular to) may be generated by s p oints h for each of which \h \ = delta = d + 1. The underlying theory, whic h suggests that such a restriction might be helpful, is presented. The gene ral character of the search is described, and, for s = 3, d less than or eq ual to 29 and s = 4, d less than or equal to 23, a list of K-optimal rules is given. It is not known whether these are also optimal rules in the gener al sense; this matter is discussed.