Perfect Quadratic Forms -- an Upper Bound and Challenges in Enumeration

In 1908 Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must correspond to a so- called perfect (quadratic) form and his algorithm enumerates the finitely many perfect forms up to similarity in a fixed dimension. However, the number of non-similar perfect forms and the comlexity of the algorithm grows quickly in the dimension and as a result Voronoi’s algorithm has only been completely executed up to dimension 8. We discuss an upper bound on the number of perfect forms and the challenges that arise for completing Voronoi's algorithm in dimension 9.