Weak isometries of the Boolean cube

Consider the metric space C consisting of the n-dimensional Boolean cube equipped with the Hamming distance. A weak isometry of C is a permutation of C preserving a given subset of Hamming distances. In Krasin (2006), Krasin showed that in most cases preserving a single Hamming distance forces a weak isometry to be an isometry. In this article we study those weak isometries that are not automatically an isometry, providing a complete classification of weak isometries of C.