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

Abstract

Let Fq be a finite field, where q is an odd prime power. Let R=Fq+uFq+vFq+uvFq 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 the 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. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.