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

TL;DR

This paper investigates the high-density behavior of Hamming codes modeled as hard spheres on a lattice, revealing potential phase transitions and discussing implications for coding theory and density bounds.

Contribution

It provides a solution to the liquid phase equations at high densities and conjectures a phase transition, connecting statistical physics with coding theory.

Findings

Liquid phase solution valid at large densities

Negative entropy indicates phase transition

Discussion of bounds on maximal density

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.