Challenges in Enumerating Perfect Quadratic Forms

Abstract

We consider the lattice packing problem and a classical method to solve it by enumerating perfect quadratic forms. We show an improved upper bound on the number of non-similar perfect forms based on a volumetric argument and lattice reduction theory. Furthermore we take a look at the challenges that arise when enumerating 9-dimensional perfect forms and we try to approximate the number of them using theoretical and computational results.