We introduce 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 $G$. For a pair of proper $q$-colorings $X, Y$ of $G$, we define their distance as the minimum Hamming distance between $X$ and $\sigma(Y)$ over all $\sigma \in S_q$. We then say that a set of proper $q$-colorings of $G$ is $\delta$-distinct if any pair of colorings in the set have distance at least $\delta n$. We investigate how one-sided spectral expansion relates to the largest possible set of $\delta$-distinct 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 maximal size of a set of $\delta$-distinct colorings of any $d$-regular graph on at most $n$ vertices with normalized second eigenvalue at most $\lambda$. We study the growth of $f$ as $n$ goes to infinity. We partially characterize regimes of $(\delta, \lambda)$ where $f$ grows exponentially, is finite, and is at most $1$, respectively. We also prove several sharp phase transitions between these regimes.

翻译：我们引入了一种新的$[q]^n$上的纠错码概念，其中码是某个固定$n$顶点图$G$的一组真$q$-着色。对于图$G$的一对真$q$-着色$X, Y$，我们定义它们的距离为$X$与$\sigma(Y)$在所有$\sigma \in S_q$上的最小汉明距离。如果一组$G$的真$q$-着色中任意一对着色之间的距离至少为$\delta n$，则我们称该组着色是$\delta$-可区分的。我们研究了单侧谱扩展如何与图上最大可能的$\delta$-可区分着色集合相关联。对于固定的$(\delta, \lambda) \in [0, 1] \times [-1, 1]$和正整数$d$，令$f_{\delta, \lambda, d}(n)$表示在任何顶点数不超过$n$、归一化第二特征值至多为$\lambda$的$d$-正则图上，$\delta$-可区分着色集合的最大规模。我们研究了当$n$趋于无穷时$f$的增长情况。我们部分刻画了$(\delta, \lambda)$在哪些区域中$f$呈指数增长、有限以及至多为$1$。我们还证明了这些区域之间几个尖锐的相变。