Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68

We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over \begin{document}$ \mathbb{F}_4 $\end{document} . These codes have binary images with parameters \begin{document}$ [32,16,8] $\end{document} or \begin{document}$ [32,16,6] $\end{document} . These are lifted to codes over \begin{document}$ \mathbb{F}_4+u\mathbb{F}_4 $\end{document} , to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over \begin{document}$ \mathbb{F}_2+u\mathbb{F}_2 $\end{document} to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.