MATHEMATICS OF COMPUTATION VOLUME 53, NUMBER 188 OCTOBER 1989, PAGES 627-637 Some Properties of Rank-2 Lattice Rules* By J. N. Lyness and I. H. Sloan Abstract. A rank-2 lattice rule is a quadrature rule for the (unit) s-dimensional hy- percube, of the form "1 "2 Qf = (l/nin2) 22 z2 ^X«l/»l +32*2/112), Jl=l J2 = l which cannot be re-expressed in an analogous form with a single sum. Here / is a periodic extension of /, and z-i, z2 are integer vectors. In this paper we discuss these rules in detail; in particular, we categorize a special subclass, whose leading one- and two-dimensional projections contain the maximum feasible number of abscissas. We show that rules of this subclass can be expressed uniquely in a simple tricycle form. 1. Introduction. 1.1. Background to Lattice Rules. Lattice rules are numerical quadrature rules for integration over an s-dimensional hypercube. They are generalizations of the one- dimensional trapezoidal rule which employ abscissas that lie on an s-dimensional lattice. A well-known and important subclass of lattice rules are the number- theoretic rules of Korobov [7]. There is a large literature devoted to number- theoretic rules, some of which appears in the reference list. Lattice rules were first explicitly introduced by Sloan [10] and Sloan and Ka- choyan [11]. In terms of an s-dimensional integration lattice L which contains the integer lattice Zs, the corresponding lattice rule is defined by (Li) QLf = ^k) S /M. where A(Ql) is the set of lattice points contained within the half-open unit cube of integration, and u(Ql) is the number of such points. Here / is a periodic continuation of /. In Sloan and Kachoyan [11], many properties of lattice rules were derived, based on definition (1.1) and under the assumption that / is continuous. The theory was developed further in Sloan and Lyness [12], exploiting the more convenient definition (1.2) below. It is almost obvious that when t and n¿ are positive integers and the components of z¿ = (z\,z2,..., zf) are integers, the form 1 ni n2 n( / v (1.2) Qf=—— ££■■■£/(*-+*.- + ■•■+*-) Jl=lj2 = l 3t = l Received July 6, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 65D32. "This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U. S. Department of Energy, under contract W-31-109-Eng-38. ©1989 American Mathematical Society 0025-5718/89 $1.00 + $.25 per page 627 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use