Lattices (discrete subgroups of n-dimensional Euclidean spaces) are ubiquitous objects in mathematics.

In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. The Voronoi cell of a zonotopal lattice is a zonotope, i.e., a projection of a regular cube. Examples of zonotopal lattices include lattices of Voronoi's first kind and tensor products of root lattices of type $\mathsf{A}$. The combinatorial structure of zonotopal lattices can be described by regular matroids/totally unimodular matrices. We observe that a linear algebra version of the minimum mean cycle canceling method can be applied for efficiently solving the closest vector problem in a zonotopal lattice if the lattice is given as the integral kernel of a totally unimodular matrix.

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.

This paper develops a new method to obtain upper bounds for spherical codes, based on semidefinite programming. With this method we improve the previous bounds for the kissing number in several dimensions, as well as other classical problems like Tammes' problem.