On \((\theta , \Theta )\)-cyclic codes and their applications in constructing QECCs

Let \({\mathbb {F}}_q\) be a finite field, where q is an odd prime power. Let \(R={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q+uv{\mathbb {F}}_q\) with \(u^2=u,v^2=v,uv=vu\). In this paper, we study the algebraic structure of \((\theta , \Theta )\)-cyclic codes of block length (r, s) over \({\mathbb {F}}_qR.\) Specifically, we analyze the structure of these codes as left \(R[x:\Theta ]\)-submodules of \({\mathfrak {R}}_{r,s} = \frac{{\mathbb {F}}_q[x:\theta ]}{\langle x^r-1\rangle } \times \frac{R[x:\Theta ]}{\langle x^s-1\rangle }\). 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 \((\theta , \Theta )\)-cyclic codes over \({\mathbb {F}}_qR\) and their duals is established. Moreover, we calculate the generator polynomials of the dual of \((\theta , \Theta )\)-cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from \((\theta , \Theta )\)-cyclic codes of block length (r, s) over \({\mathbb {F}}_qR\). We support our theoretical results with illustrative examples.