On the perfectness of minimal regular partitions of the edge set of the $n$-dimensional cube

We prove that, for $n$ equal to $3$, $5$, and a power of $2$, every minimal partition of the edge set of the $n$-dimensional cube is perfect. As a consequence, we obtain some description of the classes of all minimal parallel-serial contact schemes ($\pi$-schemes) realizing the linear Boolean functions that depend essentially on $n$ variables for the corresponding values of $n$. Bibliogr. 16.