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

Let Fq be a finite field, where q is an odd prime power. Let R=Fq+uFq+vFq+uv Fq with u²=u,v²=v,uv=vu. In this paper, we study the algebraic structure of (θ, Θ)-cyclic codes of block length $(r,s )$ over FqR. Specifically, we analyze the structure of these codes as left R[x:Θ]-submodules of Rr,s = Fq[x:θ]/ xʳ-1 × R[x:Θ]/ xˢ-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 FqR 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 FqR. We support our theoretical results with illustrative examples.