Self-Dual Skew Cyclic Codes over $mathbb{F}_{q}+umathbb{F}_{q}$

In this paper, we give conditions for the existence of Hermitian self-dual $Theta-$cyclic and $Theta-$negacyclic codes over the finite chain ring $mathbb{F}_q+umathbb{F}_q$. By defining a Gray map from $R=mathbb{F}_q+umathbb{F}_q$ to $mathbb{F}_{q}^{2}$, we prove that the Gray images of skew cyclic codes of odd length $n$ over $R$ with even characteristic are equivalent to skew quasi-twisted codes of length $2n$ over $mathbb{F}_q$ of index $2$. We also extend an algorithm of Boucher and Ulmer cite{BF3} to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over $mathbb{F}_q+umathbb{F}_q$.