Group codes form an important class of spherical codes, consisting of a single orbit under a subgroup of the orthogonal group. They have been studied by (Mittelholzer-Lahtonen, 1996) in the case of Coxeter groups for their packing radius. We study them here for the same groups, with respect to their covering radius. An exact algorithm to determine their covering radii is derived, based on geometric ideas, and applied to tabulate their values in dimensions up to 8. The values of the covering radii are compared to the sphere covering bound, and to the bounds in (Fazekas-Levenshtein, 1995) and (Boyvalenkov-Stoyanova, 2021) on the covering radius of spherical designs with given strength. For many sizes of codes, our codes are reasonably sparse sphere coverings, and the only known in these dimensions.

In a recent paper, Garber, Gavrilyuk, and Magazinov [ Discrete Comput. Geom., 53 (2015), pp. 245--260] proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all 5-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in $\mathbb{R}^5$ holds if and only if every 5-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that $P$ is combinatorially equivalent to a Dirichlet--Voronoi polytope for some lattice $\Lambda$, and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of $P$ with the Voronoi tiling of $\Lambda$. We also propose a new condition which, if satisfied by a parallelohedron $P$, is sufficient to infer that $P$ is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron and cohomologies of this complex.

A polyhedral norm is a norm N on R^n for which the set N(x)\leq 1 is a polytope. This covers the case of the L^1 and L^{\infty} norms. We consider here effective algorithms for determining the Voronoi polytope for such norms with a point set being a lattice. The algorithms, that we propose, use the symmetries effectively in order to compute a decomposition of the space into convex polytopes named {\em $VN$-spaces}. The Voronoi polytopes and other geometrical information are easily obtained from it.

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.