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Cyclic type factorizations of complete bipartite graphs into hypercubes

So far, the smallest complete bipartite graph which was known to have a cyclic type decomposition into cubes Qd of a given dimension d was Kd2d−2,d2d−2. Using binary Hamming codes we prove in this paper that there exists a cyclic type factorization of K2d−1,2d−1 into Qd if and only if d is a power of 2.