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

Let $\mathbb Fq$ be a finite field, where $q$ is an odd prime power. Let $R=\mathbb{F}q+u\mathbb{F}q+v\mathbb{F}q+uv\mathbb Fq$ with $u^{2=u,v2=v,uv=vu$.} 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 $\mathbb FqR$ 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.