Sphere packing proper colorings of an expander graph

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.