  A cube-like graph is a Cayley graph for the elementary abelian group of order
$2^n$. In studies of the chromatic number of cube-like graphs, the $k$th power
of the $n$-dimensional hypercube, $Q_n^k$, is frequently considered. This
coloring problem can be considered in the framework of coding theory, as the
graph $Q_n^k$ can be constructed with one vertex for each binary word of length
$n$ and edges between vertices exactly when the Hamming distance between the
corresponding words is at most $k$. Consequently, a proper coloring of $Q_n^k$
corresponds to a partition of the $n$-dimensional binary Hamming space into
codes with minimum distance at least $k+1$. The smallest open case, the
chromatic number of $Q_8^2$, is here settled by finding a 13-coloring. Such
13-colorings with specific symmetries are further classified.
