In this paper, we construct new self-dual codes from a construction that involves a unique combination; \begin{document}$ 2 \times 2 $\end{document} block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings \begin{document}$ \mathbb{F}_2 $\end{document}, \begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document} and \begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document}. Using extensions and neighbours of codes, we construct \begin{document}$ 32 $\end{document} new self-dual codes of length \begin{document}$ 68 $\end{document}. We construct 48 new best known singly-even self-dual codes of length 96.
