Lattice packings: an upper bound on the number of perfect lattices

Abstract

We consider an algorithmic approach by Voronoi to solve the lattice packing problem in any fixed dimension. After a visual introduction that makes clear why we should be interested in enumerating so-called perfect lattices, we continue with proving an upper bound on the number of them. This proof will mostly use geometric properties of perfect lattices and we conclude by a volumetric argument.