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

Cruz  Mohan; Karthick Gowdhaman; Gokul  Radhakrishnan; Durairajan  Chinnapillai; Irfan  Siap

doi:10.13069/jacodesmath.v13i2.453

Cruz Mohan Department of Mathematics, Bishop Heber College, Tiruchirappalli - 620 017, Tamil Nadu, India
Karthick Gowdhaman Department of Mathematics, SASTRA Deemed University, Thanjavur-613401, Tamil Nadu, India
Gokul Radhakrishnan Department of Mathematics, Bharathidasan University, Tiruchirappalli - 620 024, Tamil Nadu, India
Durairajan Chinnapillai Durairajan Chinnapillai;Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamil Nadu, India
Irfan Siap Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia

DOI: https://doi.org/10.13069/jacodesmath.v13i2.453

Keywords: Linear codes, $\Theta_t$-cyclic codes, Non-chain rings, Gray maps, $(\Theta_t,\lambda)$-Constacyclic codes, DNA-codes

Abstract

This article studies Θ_t-cyclic and (Θ_t, λ)-constacyclic codes over the finite commutative non-chain Frobenius ring R = F_q[u, 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.