The determination of canonical forms for lattice quadrature rules

Lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s-dimensional unit hypercube Us = [0, 1)s. For a given lattice rule, say QL, a set of points L (the integration lattice), regularly spaced in all of Rs, is generated by a finite number of rational vectors. The abscissa set for QL is then P(QL)= L ∩ Us. It is known that P(QL) is a finite Abelian group under addition modulo the integer lattice Zs, and that QL(f) may be written in the form of a nonrepetitive multiple sum, QL(f)=1n1⋯nm∑j1=1n1⋯∑jm=1nmfj1n1z1+⋯+jmnmzm, known as a canonical form, in which + denotes addition modulo Zs. In this form, zi ∈ Zs, m is called the rank and n1, n2,…, nm are called the invariants of QL, and ni+1¦ni for i = 1,2,…, m − 1. The rank and invariants are uniquely determined for a given lattice rule. In this paper we provide a construction of a canonical form for a lattice rule QL, given a generator set for the lattice L. We then show how the rank and invariants of QL may be determined directly from the generators of the dual lattice L⊥.