New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 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}