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

TL;DR

This paper determines the minimum number of colors needed to color the 8-dimensional hypercube's square so that no two vertices of the same color are close, connecting it to binary codes with certain minimum distances.

Contribution

It provides an explicit coloring of the 8-cube's square, establishing that 13 colors suffice, and links this to the minimum number of binary codes with specified minimum distance.

Findings

 colors are sufficient for the 8-cube's square

Explicit coloring construction provided

Connects coloring problem to binary code partitioning

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 $\chi_{\bar{2}}(8)=13$.