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 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.

翻译：设$\mathbb F_q$为有限域，其中$q$为奇素数幂。令$R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$，满足$u^2=u,v^2=v,uv=vu$。本文研究$\mathbb{F}_qR$上块长$(r,s)$的$(\theta, \Theta)$-循环码的代数结构。具体地，我们将此类码作为左$R[x:\Theta]$-子模在$\mathfrak{R}_{r,s} = \frac{\mathbb{F}_q[x:\theta]}{\langle x^r-1\rangle} \times \frac{R[x:\Theta]}{\langle x^s-1\rangle}$中的结构进行分析。研究涉及确定这类码的生成多项式与最小生成集。进一步，我们讨论了可分码的代数结构，建立了$\mathbb F_qR$上$(\theta, \Theta)$-循环码与其对偶码生成多项式之间的关系，并计算了对偶码的生成多项式。作为应用，我们利用$\mathbb{F}_qR$上块长$(r,s)$的$(\theta, \Theta)$-循环码构造了量子纠错码（QECCs）。最后通过实例验证理论结果。