On the high density behavior of Hamming codes with fixed minimum distance

Abstract

We discuss the high density behavior of a system of hard spheres of diameter d on the hypercubic lattice of dimension n, in the limit n -> oo, d -> oo, d/n=delta. The problem is relevant for coding theory. We find a solution to the equations describing the liquid up to very large values of the density, but we show that this solution gives a negative entropy for the liquid phase when the density is large enough. We then conjecture that a phase transition towards a different phase might take place, and we discuss possible scenarios for this transition. Finally we discuss the relation between our results and known rigorous bounds on the maximal density of the system.