Symmetric vertex partitions of hypercubes by isometric trees (Q913820)

For \(n=2^{k-1}\) the author proves, via a counting argument, that for any tree \({\mathcal T}\) with vertex set T, if \({\mathcal T}\) is isometrically embedded in the n-cube \(Q_ n\), then there exists a subgroup G of \(Aut(Q_ n)\) such that \(\{\) g(T)\(| g\in G\}\) is a vertex partition of \(Q_ n\). This generalizes the theorem of Hamming on the existence of perfect single-error-correcting codes, which corresponds to the case where \({\mathcal T}\) is an n-star. For the special case of an antipodal path the author gives an explicit construction of the group G, thereby answering a question of D. Rogers.