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

$\Theta_S-$cyclic codes over $A_k$