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.
Additional Metadata
Keywords lattice, Voronoi cell, Delone cell, covering radius, quantizer constant, lattice isomorphism problem, zonotope
MSC Complexity of computation (including implicit computational complexity) (msc 03D15)
THEME Logistics (theme 3)
Publisher A.M.S.
Journal Mathematics of Computation
Citation
Dutour Sikiric, M, Schuermann, A, & Vallentin, F. (2009). Complexity and algorithms for computing Voronoi cells of lattices. Mathematics of Computation, 78, 1713–1731.