On the traveling salesman problem in binary hamming spaces

Given a subset X of vertices of the n-cube (i.e., the n-dimensional Hamming space), we are interested in the solution of the traveling salesman problem; namely, the minimal length of a cycle passing through all vertices of X. For a given number M, we estimate the maximum of these lengths when X ranges over all possible choices of sets of M vertices. Asymptotically, our estimates show that for a number M of vertices growing exponentially in n, the maximum is attained for a code with maximal possible minimum distance.