Note on cyclic decompositions of complete bipartite graphs into cubes

So far, the smallest complete bipartite graph which was known to have a cyclic decomposition into cubes ${\displaystyle Q_d}$ of a given dimension d was ${\displaystyle K_{d{2}^{d-1},d{2}^{d-2}}}$. We improve this result and show that also ${\displaystyle K_{d{2}^{d-2},d{2}^{d-2}}}$ allows a cyclic decomposition into ${\displaystyle Q_d}$. We also present a cyclic factorization of ${\displaystyle K_{8,8}}$ into Q₄.