Exact calculation of quantizer constants for arbitrary lattices

Abstract

We present an algorithm for the exact computer-aided construction of the Voronoi cells of lattices with known symmetry group. Our algorithm scales better than linearly with the total number of faces and is applicable to dimensions beyond 12, which previous methods could not achieve. The new algorithm is applied to the Coxeter-Todd lattice $K_{12}$ as well as to a family of lattices obtained from laminating $K_{12}$. By optimizing this family, we obtain a new 13-dimensional lattice, whose quantizer constant is smaller than any published at the time of submission. (For subsequent improvements, see Note added in proof after the Conclusions.)