ABSTRACT. In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a # P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.
In REF it is also shown that computing the number of vertices of the Voronoi cell of a lattice is #P -hard.
Complexity and algorithms for computing Voronoi cells of lattices
Math. Comp. 267 (2009), 1713-1731
Mathematics Computer Science