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

Abstract

<p style='text-indent:20px;'>A binary code <inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{C}} $\end{document}</tex-math></inline-formula> of length <inline-formula><tex-math id="M3">\begin{document}$ n = \sum_{i = 1}^{m}n_i $\end{document}</tex-math></inline-formula> and minimum distance <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula> is said to be of multiply constant-weight and denoted by MCWC<inline-formula><tex-math id="M5">\begin{document}$ (w_1, n_1 $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M6">\begin{document}$ w_2, n_2 $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M7">\begin{document}$ \ldots $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M8">\begin{document}$ w_m, n_m $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M9">\begin{document}$ d) $\end{document}</tex-math></inline-formula>, if each codeword has weight <inline-formula><tex-math id="M10">\begin{document}$ w_1 $\end{document}</tex-math></inline-formula> in the first <inline-formula><tex-math id="M11">\begin{document}$ n_1 $\end{document}</tex-math></inline-formula> coordinates, weight <inline-formula><tex-math id="M12">\begin{document}$ w_2 $\end{document}</tex-math></inline-formula> in the next <inline-formula><tex-math id="M13">\begin{document}$ n_2 $\end{document}</tex-math></inline-formula> coordinates, and so on and so forth. Multiply constant-weight codes (MCWCs) can be utilized to improve the reliability of certain physically unclonable function response and has been widely studied. Research showed that multiply constant-weight codes are equivalent to generalized packing designs and generalized Howell designs (GHDs) can be regarded as generalized packing designs with a special block type. In this paper, we give combinatorial constructions for optimal MCWC<inline-formula><tex-math id="M14">\begin{document}$ (3, n_1;1, n_2;1, n_3;8) $\end{document}</tex-math></inline-formula>s by a class of generalized packing designs, which come from generalized Howell designs. Furthermore, for <inline-formula><tex-math id="M15">\begin{document}$ e = 3, 4, 5 $\end{document}</tex-math></inline-formula>, we prove that there exists a GHD <inline-formula><tex-math id="M16">\begin{document}$ (n+e, 3n) $\end{document}</tex-math></inline-formula> if and only if <inline-formula><tex-math id="M17">\begin{document}$ n\ge 2e+1 $\end{document}</tex-math></inline-formula> leaving several possibly exceptions.</p>