Self-Dual Skew Cyclic Codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q$

In this paper, we give conditions for the existence of Hermitian self-dual $\mathrm{\Theta }-$cyclic and $\mathrm{\Theta }-$negacyclic codes over the finite chain ring ${\mathbb{F}}_q+u{\mathbb{F}}_q$. By defining a Gray map from $R={\mathbb{F}}_q+u{\mathbb{F}}_q$ 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 ${\mathbb{F}}_q+u{\mathbb{F}}_q$.