Sphere packing proper colorings of an expander graph

TL;DR

This paper introduces graphical error-correcting codes based on proper colorings of graphs and explores how spectral expansion properties influence the maximum size of such codes, revealing phase transitions in their growth.

Contribution

It defines a new class of error-correcting codes on graphs using proper colorings and analyzes their size relative to spectral expansion parameters.

Findings

Characterized regimes where code size grows exponentially or remains bounded.

Proved phase transitions between different growth regimes.

Connected spectral properties of graphs to error-correcting code capacities.

Abstract

We introduce graphical error-correcting codes, a new notion of error-correcting codes on $[q]^n$, where a code is a set of proper $q$-colorings of some fixed $n$-vertex graph $G$. We then say that a set of $M$ proper $q$-colorings of $G$ form a $(G, M, d)$ code if any pair of colorings in the set have Hamming distance at least $d$. This directly generalizes typical $(n, M, d)$ codes of $q$-ary strings of length $n$ since we can take $G$ as the empty graph on $n$ vertices. We investigate how one-sided spectral expansion relates to the largest possible set of error-correcting colorings on a graph. For fixed $(\delta, \lambda) \in [0, 1] \times [-1, 1]$ and positive integer $d$, let $f_{\delta, \lambda, d}(n)$ denote the maximum $M$ such that there exists some $d$-regular graph $G$ on at most $n$ vertices with normalized second eigenvalue at most $\lambda$ that has a $(G, M, d)$ code. We study the growth of $f$ as $n$ goes to infinity. We partially characterize the regimes of $(\delta, \lambda)$ where $f$ grows exponentially or is bounded by a constant, respectively. We also prove several sharp phase transitions between these regimes.