On (θ, Θ)-cyclic codes and their applications in constructing QECCs

Let 𝔽_q be a finite field, where q is an odd prime power. Let R=𝔽_q+u𝔽_q+v𝔽_q+uv𝔽_q with u^2=u,v^2=v,uv=vu. In this paper, we study the algebraic structure of (θ, Θ)-cyclic codes of block length (r,s ) over 𝔽_qR. Specifically, we analyze the structure of these codes as left R[x:Θ]-submodules of ℜ_r,s = 𝔽_q[x:θ]/⟨ x^r-1⟩×R[x:Θ]/⟨ x^s-1⟩. Our investigation involves determining generator polynomials and minimal generating sets for this family of codes. Further, we discuss the algebraic structure of separable codes. A relationship between the generator polynomials of (θ, Θ)-cyclic codes over 𝔽_qR and their duals is established. Moreover, we calculate the generator polynomials of dual of (θ, Θ)-cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from (θ, Θ)-cyclic codes of block length (r,s) over 𝔽_qR. We support our theoretical results with illustrative examples.