${\mathrm{\Theta }}_S-$cyclic codes over $A_k$

We study ${\mathrm{\Theta }}_S-$cyclic codes over the family of rings $A_k\mathrm{.}$ We characterize ${\mathrm{\Theta }}_S-$cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is ${\mathrm{\Theta }}_S-$cyclic then its Hermitian dual is also ${\mathrm{\Theta }}_S-$cyclic. Finally, we give constructions of ${\mathrm{\Theta }}_S-$cyclic codes.