  A multidimensional nonnegative matrix is called polystochastic if the sum of
entries in each line is equal to $1$. The set of all polystochastic matrices of
order $n$ and dimension $d$ is a convex polytope $\Omega_n^d$. In the present
paper, we compare known bounds on the number of vertices of the polytope
$\Omega_n^d$ and prove that the number of vertices of $\Omega_3^d$ is doubly
exponential on $d$.
