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 d over the hypercube [ 0 , 1 ) s [0,1)^s has been undertaken. The search is restricted to a population K ( s , δ ) K(s,\delta ) of lattice rules Q ( Λ ) Q(\Lambda ) . This includes those where the dual lattice Λ ⊥ \Lambda ^\perp may be generated by s s points h \bf h for each of which | h | = δ = d + 1 |\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 s=3 , d ≤ 29 d \leq 29 and s = 4 s=4 , d ≤ 23 d \leq 23 , a list of K K -optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.