Cavity approach to sphere packing in Hamming space

In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of Gilbert and Varshamov. Consistently with known numerical results, the replica symmetric equations also suggest a crystalline solution, where for even diameters the spheres are more likely to be found in one of the subspaces (even or odd) of the Hamming space. These crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultrametric space. Finally, we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres. Known maximum packings are reproduced efficiently in nontrivial ranges of dimensions and number of spheres.