New type i binary [72, 36, 12] self-dual codes from composite matrices and <i>R</i>₁ lifts

In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form \begin{document}$ [I_n \ | \ \Omega(v)], $\end{document} where \begin{document}$ I_n $\end{document} is the identity matrix and \begin{document}$ \Omega(v) $\end{document} is a composite matrix and search for binary self-dual codes with parameters \begin{document}$ [36,18, 6 \ \text{or} \ 8]. $\end{document} We next lift these codes over the ring \begin{document}$ R_1 = \mathbb{F}_2+u\mathbb{F}_2 $\end{document} to obtain codes whose binary images are self-dual codes with parameters \begin{document}$ [72,36,12]. $\end{document} Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find \begin{document}$ 30 $\end{document} new Type I binary self-dual codes with parameters \begin{document}$ [72,36,12]. $\end{document}