Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $

We examine the binary codes from adjacency matrices of the graph with vertices the nodes of the \begin{document}$ m $\end{document} -ary \begin{document}$ n $\end{document} -cube \begin{document}$ Q^m_n $\end{document} and with adjacency defined by the Lee metric. For \begin{document}$ n = 2 $\end{document} and \begin{document}$ m $\end{document} odd, we obtain the parameters of the code and its dual, and show the codes to be \begin{document}$ LCD $\end{document} . We also find \begin{document}$ s $\end{document} -PD-sets of size \begin{document}$ s+1 $\end{document} for \begin{document}$ s for the dual codes, i.e. \begin{document}$ [m^2,2m-1,m]_2 $\end{document} codes, when \begin{document}$ n = 2 $\end{document} and \begin{document}$ m\ge 5 $\end{document} is odd.