Vertices of the polytope of polystochastic matrices and product constructions

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries at each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope ${\mathrm{\Omega }}_n^d$. In the present paper, we compare known bounds on the number $V(n,d)$ of vertices of the polytope ${\mathrm{\Omega }}_n^d$, propose two constructions of vertices of ${\mathrm{\Omega }}_n^d$ based on multidimensional matrix multiplication, and list all vertices of the polytope ${\mathrm{\Omega }}_3^4$.