On the number of partitions of the hypercube $Z_q^n$ into large subcubes

It is proved that the number of partitions of the hypercube $Z_q^n$ into $q^m$ subcubes of dimension $n-m$ each for fixed $q$, $m$ and growing $n$ is asymptotically equal to $n^{(q^m-1)/(q-1)}$.