Structure of \({\mathbb {F}}_q{\mathcal {R}}\)-linear \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes

Let \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ be the finite field of order \(q=p^m\) $q=p^m$ where p is a prime, and m is a positive integer. This work introduces \({\mathbb {F}}_q{\mathcal {R}}\) ${\mathbb{F}}_q\mathcal{R}$ -linear \((\varTheta ,\varDelta _\varTheta )\) $(\Theta ,{\Delta }_{\Theta })$ -cyclic codes where \({\mathcal {R}}={\mathbb {F}}_q+u{\mathbb {F}}_q\) $\mathcal{R}={\mathbb{F}}_q+u{\mathbb{F}}_q$ with \(u^2=u\) $u^2=u$ , i.e., \((\varTheta ,\varDelta _\varTheta )\) $(\Theta ,{\Delta }_{\Theta })$ -cyclic codes over \({\mathbb {F}}_q{\mathcal {R}}\) ${\mathbb{F}}_q\mathcal{R}$ . With the help of the decomposition method, we study the structural properties and determine the generator polynomials of \({\mathbb {F}}_q{\mathcal {R}}\) ${\mathbb{F}}_q\mathcal{R}$ -linear \((\varTheta ,\varDelta _\varTheta )\) $(\Theta ,{\Delta }_{\Theta })$ -cyclic codes. Further, we define the Gray map over \({\mathbb {F}}_q{\mathcal {R}}\) ${\mathbb{F}}_q\mathcal{R}$ and find the Gray images of \({\mathbb {F}}_q{\mathcal {R}}\) ${\mathbb{F}}_q\mathcal{R}$ -linear \((\varTheta ,\varDelta _\varTheta )\) $(\Theta ,{\Delta }_{\Theta })$ -cyclic codes over \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ . Finally, with the help of our established results, we have constructed some new codes corresponding to \({\mathbb {F}}_q{\mathcal {R}}\) ${\mathbb{F}}_q\mathcal{R}$ -linear \((\varTheta ,\varDelta _\varTheta )\) $(\Theta ,{\Delta }_{\Theta })$ -cyclic codes.