A {\em packing coloring} of a graph $G$ is a mapping assigning a positive integer (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 a packing coloring of $G$ is called the {\em 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 ({\em P. Torres, M. Valencia-Pabon, The packing chromatic number of hypercubes, Discrete Appl. Math. 190--191 (2015), 127--140}) 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 packing coloring of Cartesian 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.}).

翻译：图$G$的\textbf{打包着色}是一种映射，将正整数（颜色）分配给$G$的每个顶点，使得任何两个颜色为$k$的顶点之间的距离至少为$k+1$。$G$的打包着色所需的最小颜色数称为$G$的\textbf{打包色数}。在本文中，我们继续研究超立方体的打包色数，并通过利用扩展汉明码的性质，构造距离顶点的递归子集，改进了Torres和Valencia-Pabon（{\em P. Torres, M. Valencia-Pabon, The packing chromatic number of hypercubes, Discrete Appl. Math. 190--191 (2015), 127--140}）提出的上界。我们还否定了Brešar、Klavžar和Rall（{\em Problem 5, Brešar et al., On the packing chromatic number of Cartesian products, hexagonal lattice, and trees. Discrete Appl. Math. 155 (2007), 2303--2311.}）关于笛卡尔积打包着色的一个问题。