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 grows quickly in the dimension and as a result

Voronoi’s algorithm has only been completely executed up to dimension 8.

We prove an upper bound of $\exp(O(d^^2 \log(d)))$ on the number of similarity classes [1]. The proof is mostly

geometric and concludes by a volumetric argument.

Additionally we discuss some practical challenges and progress for completing Voronoi's algorithm in dimension 9,

e.g. how to efficiently remove redundant similar quadratic forms from a large set [2].

[1] https://arxiv.org/abs/1901.04807 [2]
[2] https://arxiv.org/abs/2004.14022 [3]