A coloring of the square of the 8-cube with 13 colors

Abstract

Let ${\chi }_{\bar{k}}(n)$ be the number of colors required to color the $n$-dimensional hypercube such that no two vertices with the same color are at a distance at most $k$. In other words, ${\chi }_{\bar{k}}(n)$ is the minimum number of binary codes with minimum distance at least $k+1$ required to partition the $n$-dimensional Hamming space. By giving an explicit coloring, it is shown that ${\chi }_{\bar{2}}(8)=13$.

arXiv comment3 pages