Constructions of optimal multiply constant-weight codes MCWC$ (3,n_1;1,n_2;1,n_3;8)s $

A binary code \begin{document}$ {\mathcal{C}} $\end{document} of length \begin{document}$ n = \sum_{i = 1}^{m}n_i $\end{document} and minimum distance \begin{document}$ d $\end{document} is said to be of multiply constant-weight and denoted by MCWC \begin{document}$ (w_1, n_1 $\end{document} ; \begin{document}$ w_2, n_2 $\end{document} ; \begin{document}$ \ldots $\end{document} ; \begin{document}$ w_m, n_m $\end{document} ; \begin{document}$ d) $\end{document} , if each codeword has weight \begin{document}$ w_1 $\end{document} in the first \begin{document}$ n_1 $\end{document} coordinates, weight \begin{document}$ w_2 $\end{document} in the next \begin{document}$ n_2 $\end{document} coordinates, and so on and so forth. Multiply constant-weight codes (MCWCs) can be utilized to improve the reliability of certain physically unclonable function response and has been widely studied. Research showed that multiply constant-weight codes are equivalent to generalized packing designs and generalized Howell designs (GHDs) can be regarded as generalized packing designs with a special block type. In this paper, we give combinatorial constructions for optimal MCWC \begin{document}$ (3, n_1;1, n_2;1, n_3;8) $\end{document} s by a class of generalized packing designs, which come from generalized Howell designs. Furthermore, for \begin{document}$ e = 3, 4, 5 $\end{document} , we prove that there exists a GHD \begin{document}$ (n+e, 3n) $\end{document} if and only if \begin{document}$ n\ge 2e+1 $\end{document} leaving several possibly exceptions.