Cavity approach to sphere packing in Hamming space

TL;DR

This paper applies the cavity method to analyze sphere packing in Hamming space, revealing asymptotic optimality of certain bounds, crystalline structures, and efficient algorithms for dense packings.

Contribution

It introduces a cavity method approach to sphere packing in Hamming space, demonstrating asymptotic optimality and proposing algorithms for dense packings.

Findings

Replica symmetric and symmetry breaking approximations match Gilbert-Varshamov bounds.

Crystalline solutions emerge for even diameters, with spheres favoring subspaces.

Algorithms efficiently reproduce known maximum packings in various dimensions.

Abstract

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 ultra-metric 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 non trivial ranges of dimensions and number of spheres.