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

Abstract:

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 $K(s,\delta )$ of lattice rules $Q(\Lambda )$. This includes those where the dual lattice $\Lambda ^\perp$ may be generated by $s$ points $\bf h$ for each of which $|\textbf {h} | = \delta =d+1$. The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for $s=3$, $d \leq 29$ and $s=4$, $d \leq 23$, 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.