by Mathieu Dutour Sikiric and Achill Schürmann and Frank Vallentin.

**Series:** 2008-19, Preprints

- MSC:
- 03D15 Complexity of computation (including implicit computational complexity)
- 11H56 Automorphism groups of lattices
- 11H06 Lattices and convex bodies
- 52B12 Special polytopes (linear programming, centrally symmetric, etc.)

**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 \sharpp-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.

**Keywords:**

lattice, Voronoi cell, Delone cell, covering radius, quantizer constant, lattice isomorphism problem, zonotope