mathematics of computation volume 54,number 190 april 1990,pages 649-660 LATTICE RULES: PROJECTION REGULARITY AND UNIQUE REPRESENTATIONS I. H. SLOANAND J. N. LYNESS Abstract. We introduce a unique characterization for lattice rules which are projection regular. Any such rule, having invariants nl, n2, ... , ns, may be expressed, uniquely, in the form T where the matrix Z = (z, , z2 , ... , zs) is upper unit triangular and individual elements satisfy 0 < z'c) < {nr/nc), r < c . 1. Introduction The notion of a lattice rule for numerical integration over the unit s-dimen- sional cube was introduced in Sloan [4] and Sloan and Kachoyan [5], and further discussed in Sloan and Lyness [6], where it was shown that any 5-dimensional lattice rule Qs can be expressed in the canonical form ('■') e^^^EE-E/Ef • 1 2 * 7i = '72=l Á=l \/=l ' / Here the invariants n,, «2, ... , ns are positive integers satisfying (1.2) ",+iK> / = 1,...,j-1; zi;e Zs for i = I, ... , s; and / is a 1-periodic extension of /. (For a more precise specification of f see Sloan and Lyness [6].) The abscissa set of the rule Qs is the set (1.3) ^(Oí) = ||¿^|:;/ = 1, -..,",, *=1,...,*}, where {v} denotes the vector whose components are the fractional parts of those of v. Clearly, {v} € [0, 1 )s, the half-open unit cube. The order of Qs is Received March 27, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 65D30. 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, and in part by the Australian Research Council. © 1990 American Mathematical Society 0025-5718/90 $1.00+ $.25 per page 649 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use