A Coloring Of The Square Of The 8-cube With 13 Colors

Kokkala Janne I., Östergård Patric R. J.. Arxiv 2015

[Paper]    
ARXIV

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

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