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

Abstract: Let $\mathbb F_q$ be a finite field, where $q$ is an odd prime power. Let $$R=\mathbb{F}<em>q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$$ with $u<sup>2=u,v<sup>2=v,uv=vu$. 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 $$\mathfrak{R}</em>{r,s} = \frac{\mathbb{F}_q[x:\theta]}{\langle x^r-1\rangle} \times \frac{R[x:\Theta]}{\langle x<sup>s-1\rangle}$$. 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 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.