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 ${n_1},{n_2}, \ldots ,{n_s}$. The number $\nu (Q)$ of abscissas required by the rule is the product ${n_1}{n_2} \cdots {n_s}$, 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 ${n^s}$ points is a 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.