The representation of lattice quadrature rules as multiple sums

We provide a classification of lattice rules. Applying elementary group theory, we assign to each s-dimensional lattice rule a rank m and a set of positive integer invariants n/sub 1/ n/sub 2/....,n/sub s/. The number ..nu..(Q) of abscissas required by the rule is the product n/sub 1/n/sub 2/...n/sub s/, and the rule may be expressed in a canonical form with m independent summations. Under this classification an N-point number-theoretic rule in the sense of Korobov and Conroy is a rank m = 1 rule having invariants N, 1,1,...,1, and the product trapezoidal rule using n/sup s/ points in rank m = s rule having invariants n,n...,n. Besides providing a canonical form, we give some of the properties of copy rules and of projections into lower dimensions.