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The packing chromatic number $\chi_\rho(G)$ of a graph $G$ is the smallestinteger $k$ needed to proper color the vertices of $G$ in such a way thedistance between any two vertices having color $i$ be at least $i+1$. We obtain$\chi_\rho(H_{q,m})$ for $m=3$, where $H_{q,m}$ is the Hamming graph of wordsof length $m$ and alphabet with $q$ symbols, and tabulate bounds of them for $m\geq 4$ up to 10000 vertices. We also give a polynomial reduction from theproblem of finding $\chi_\rho(G)$ to the Maximum Stable Set problem.

On the Packing Chromatic Number on Hamming Graphs and General Graphs