Packing coloring of hypercubes with extended Hamming codes

A packing $k$-coloring of a graph $G$ is a mapping assigning a positive integer (a color) from the set $\left(1,\dots ,k\right)$ to every vertex of $G$ such that every two distinct vertices of color $c$ are at distance at least $c+1$. The minimum value $k$ such that $G$ admits a packing $k$-coloring is called the packing chromatic number of $G$. In this paper, we continue the study of the packing chromatic number of hypercubes and we improve the upper bounds reported by Torres and Valencia-Pabon (2015) by presenting recursive constructions of subsets of distant vertices making use of the properties of the extended Hamming codes. We also answer in negative a question on the packing chromatic number of Cartesian products raised by Brešar et al. (2007).