On hamming hypercubes

Question

Regularly in a Hamming hypercube, the vertices are labelled so that edge difference (minimum number of edges traversed between two vertices) equals Hamming distance (path difference). That is lower the edge difference, lower the hamming distance(path difference) of the labelled vertices. Can we label the cube so that lower the edge difference, higher will the Hamming distance be? Not all pairs of words will satisfy this property (for instance try labeling the 2-cube). However, is there a lower bound on how many close vertices any vertex can have and still have the property? Any references for this would help. This does sound like anticodes. However I am unsure.