The Canonical Forms of a Lattice Rule
James N. Lyness · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1992
Abstract. Much of the elementary theory of lattice rules may be presented as an elegant application of classical results. These include Kronecker group representation theorem and the Hermite and Smith normal forms of integer matrices. The theory of the canonical form is a case in point. In this paper, some of this theory is treated in a constructive rather than abstract manner. A step-by-step approach that parallels the group theory is described, leading to an algorithm to obtain a canonical form of a rule of prime power order. The number of possible distinct canonical forms is derived, and this is used to determine the number of integration lattices having specied invariants. 1. Notation Used to Describe and Classify Lattice Rules An s-dimensional lattice, , is a set of points having the property that, when p and q are members of , so are p + q and p q. It may be dened by this property, together with a restriction that there are no points of accumulation. A very familiar lattice is the unit lattice 0, which comprises all points p = (p