Let M be the generator matrix of a $N$ dimensional lattice, and $V$ the set of Voronoi relevant vectors. The Voronoi cell for the origin can be written as $\text{Vor}_{\bf 0}(M)=\left\{{\bf x}: |{\bf x}|\leq|{\bf x-c}|\text{ for } \forall {\bf c}\in V\right\}$. The largest possible number of Voronoi relevant vectors for the given dimension is $2(2^N-1)$ which essentially include all the combinations of the basis vectors except the origin.
If the entries of the generator matrix M is randomly chosen, then it is almost the case that the lattice will have the largest possible number of Voronoi relevant vectors, but I would like to understand if it is possible to construct lattices with such property. The dual of the A-type root lattice is one example, according to the 3rd paragraph of this paper, but is there any other examples?
