Structure of 𝔽_qℛ -linear ( , _ ) -cyclic codes

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