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 → ∞, d → ∞, d/n = δ. 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 ≤ ρVd ≤ exp(nκ(δ)), being κ(δ) > 0 and Vd 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 ρ̃ = ρVd, but we show that this solution gives a negative entropy for the liquid phase for ρ̃ ∼ n. We then conjecture that a phase transition towards a different phase might take place, and we discuss possible scenarios for this transition.