On (, ) -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 (, ) -cyclic codes of block length (r, s) over FqR. Specifically, we analyze the structure of these codes as left Rx: -submodules of Rₑ, ₒ = Fqx: xʳ-1 Rx: 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 (, ) -cyclic codes over FqR and their duals is established. Moreover, we calculate the generator polynomials of dual of (, ) -cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from (, ) -cyclic codes of block length (r, s) over FqR. We support our theoretical results with illustrative examples.