Four-dimensional lattice rules generated by skew-circulant matrices

We introduce the class of skew-circulant lattice rules. These are $s$-dimensional lattice rules that may be generated by the rows of an $s \times s$ skew-circulant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree $d$. We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for $\delta = d+1 \leq 47,$ we have constructed an infinite sequence of rules ${\hat Q}(4,\delta)$ that has a limit rho index of $27/34 \approx 0.79$. This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.