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A {\em packing coloring} of a graph $G$ is a mapping assigning a positiveinteger (a color) to every vertex of $G$ such that every two vertices of color$k$ are at distance at least $k+1$. The least number of colors needed for apacking coloring of $G$ is called the {\em packing chromatic number} of $G$. Inthis paper, we continue the study of the packing chromatic number of hypercubesand we improve the upper bounds reported by Torres and Valencia-Pabon ({\em P.Torres, M. Valencia-Pabon, The packing chromatic number of hypercubes, DiscreteAppl. Math. 190--191 (2015), 127--140}) by presenting recursive constructionsof subsets of distant vertices making use of the properties of the extendedHamming codes. We also answer in negative a question on packing coloring ofCartesian products raised by Bre\v{s}ar, Klav\v{z}ar, and Rall ({\em Problem 5,Bre\v{s}ar et al., On the packing chromatic number of Cartesian products,hexagonal lattice, and trees. Discrete Appl. Math. 155 (2007), 2303--2311.}).

Packing coloring of hypercubes with extended Hamming codes