New self-dual and formally self-dual codes from group ring constructions

In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings \begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document} and \begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document} , we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters \begin{document}$ [72,36,14] $\end{document} .