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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.

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