Clique matchings in the $k$-ary $n$-dimensional cube

A clique matching in the $k$-ary $n$-dimensional cube (hypercube) is a collection of disjoint one-dimensional faces. A clique matching is called perfect if it covers all vertices of the hypercube. We show that the number of perfect clique matchings in the $k$-ary $n$-dimensional cube can be expressed as the $k$-dimensional permanent of the adjacency array of some hypergraph. We calculate the order of the logarithm of the number of perfect clique matchings in the $k$-ary $n$-dimensional cube for an arbitrary positive integer $k$ as $n\to\infty$.
A perfect clique matching is called precise if each two-dimensional face of the hypercube includes a sole one-dimensional face of the clique matching. Precise clique matchings are particular cases of H-designs. We prove that for the existence of precise clique matchings in the $k$-ary $n$-dimensional cube it is necessary that $k=2m$ and $n=4m$ for some positive integer m. We propose a construction of precise clique matchings for $k=2^t$ and $n=2^{t+1}$ with an arbitrary positive integer $t$.