Quaternary group ring codes: Ranks, kernels and self-dual codes

We study \begin{document}$ G $\end{document} -codes over the ring \begin{document}$ {\mathbb{Z}}_4 $\end{document} , which are codes that are held invariant by the action of an arbitrary group \begin{document}$ G $\end{document} . We view these codes as ideals in a group ring and we study the rank and kernel of these codes. We use the rank and kernel to study the image of these codes under the Gray map. We study the specific case when the group is the dihedral group and the dicyclic group. Finally, we study quaternary self-dual dihedral and dicyclic codes, tabulating the many good self-dual quaternary codes obtained via these constructions, including the octacode.