Skew cyclic codes over a finite non-chain ring and an application

This article studies Θt-cyclic and (Θt, λ)-constacyclic codes over the finite commutative non-chain Frobenius ring R = Fqu, v, w / 〈 u2 − u, v2 − v, w2 − 1, uv, uw − wu, wv − vw 〉. Gray maps, structural decompositions, and generator descriptions are developed for both odd- and even-characteristic cases. The paper further determines principal generators in the associated skew polynomial rings, dual codes, idempotent generators, and conditions for self-duality. It also presents explicit examples over specific finite fields and extends the framework to DNA codes in the even-characteristic setting through reversibility and complement constraints. Spanning sets, cardinality formulas, and optimal DNA-code constructions meeting the Griesmer bound are also obtained.