Some constructions and existence conditions for Hermitian self-dual skew codes

In this paper, we first consider the existence conditions, the construction and the enumeration of Hermitian self-dual \(\theta \) $\theta$ -cyclic and \(\theta \) $\theta$ -negacyclic codes over \(\mathrm{I\hspace{-2.10007pt}F}_{p^2}\) ${\mathrm{I}\mathrm{F}}_{p^2}$ , where p is a prime number and \(\theta \) $\theta$ is the Frobenius automorphism over \(\mathrm{I\hspace{-2.10007pt}F}_{p^2}\) ${\mathrm{I}\mathrm{F}}_{p^2}$ . We then give necessary and sufficient conditions for the existence of Hermitian self-dual \(\theta \) $\theta$ -cyclic and \(\theta \) $\theta$ -negacyclic codes over \(\mathrm{I\hspace{-2.10007pt}F}_{p^e}\) ${\mathrm{I}\mathrm{F}}_{p^e}$ where e is an even integer greater than 2.