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

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 -> infinity, d ->infinity , d/n = delta. The problem is relevant for coding theory, and the best available bounds state that the maximum density of the system falls in the interval 1 <= rho V (d) <= exp (n kappa(delta)), being kappa(delta) > 0 and V-d the volume of a sphere of radius d. We find a solution of the equations describing the liquid up to an exponentially large value of (rho) over tilde = rho V-d , but we show that this solution gives a negative entropy for the liquid phase for (rho) over tilde greater than or similar to n. We then conjecture that a phase transition towards a different phase might take place, and we discuss possible scenarios for this transition.