Equitable factorizations of Hamming shells

We construct a 1-factorization of the complement Σm of the linear Hamming code of length m=mr=2r-1 in the m-cube Qm, for r ≥ 2, having the following equitable property: its component 1-factors intersect each Cayley parallel 1-factor of Qm at a constant number of edges, (namely 2mr-r-1 edges). In the way to that construction, we find an equitable mr-1-factorization of Σm formed by two factors Ωr, Ω'r, specifically two spanning regular subgraphs, self-complementary in Σm. These results were already known for r ≤ 3, where Ω3 and Ω'3 coincide with the so-called Foster graph.