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

Source abstract

This article studies $\Theta_t$-cyclic and $(\Theta_t,\lambda)$-constacyclic codes over the finite commutative non-chain Frobenius ring $$R = \mathbb{F}_q[u,v,w]/\langle u^2-u,\ v^2-v,\ w^2-1,\ uv,\ uw-wu,\ wv-vw \rangle .$$ 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.Received: 12 September 2025 | Accepted: 05 April 2026