On the Density of a Lattice Covering for $n=11$ and $n=14$

For the Coxeter lattices $A_{11}^{4}$ and $A_{14}^{5}$, a full description of the structure of the L-partition as well as the structure of the Voronoi–Dirichlet polyhedra as polyhedra defined by their vertices is given. On the basis of this description, exact values of the covering radius and the density function are evaluated for the lattice coverings corresponding to these lattices. In both cases, the values of the density function of the covering proved to be better (less) than the formerly known values. Thus, for $n=11$ and $n=14$, improved estimates are obtained for the minimum density of lattice coverings of the Euclidean space with equal balls.