Let \(\mathbb {F}_q\) be the finite field of order \(q=p^m\), where p is a prime, m is a positive integer, and \(\mathcal {R}=\mathbb {F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle \). Thus \(\mathcal {R}[x;\theta ,\delta _\theta ]\) is a noncommutative ring, known as skew polynomial ring, where \(\theta \) is an automorphism of \(\mathcal {R}\) and \(\delta _\theta \) is a \(\theta \)-derivation of \(\mathcal {R}\). The main concern of this work is to characterize \((\theta , \delta _\theta )\)-cyclic codes over the ring \(\mathcal {R}\). Towards this, first we establish existence of the right division algorithm in \(\mathcal {R}[x;\theta ,\delta _\theta ]\). Then we find generating polynomials and idempotent generators for \((\theta , \delta _\theta )\)-cyclic codes over the ring \(\mathcal {R}\). Moreover, it is shown that \((\theta , \delta _\theta )\)-cyclic codes are principally generated. Finally, by using the decomposition method, we have provided several examples of \((\theta , \delta _\theta )\)-cyclic codes of different lengths over \(\mathcal {R}\) out of them many are optimal as per the available database (Grassl, Code Tables: bounds on the parameters of various types of codes. http://www.codetables.de/).

令\(\mathbb {F}_q\)为\(q=p^m\)阶有限域，其中p是素数，m是正整数，而\(\mathcal {R}=\mathbb {F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle \)。因此\(\mathcal {R}[x;\theta ,\delta _\theta ]\)是一个非交换环，称为偏斜多项式环，其中\(\theta \)是\(\mathcal {R }\)和\(\delta _\theta \)是\(\theta \) 的衍生\(\mathcal {R}\)。这项工作的主要关注点是表征\((\theta , \delta _\theta )\) - 环上的循环码\(\mathcal {R}\)。为此，首先我们在\(\mathcal {R}[x;\theta ,\delta _\theta ]\) 中建立右除法算法的存在。然后我们发现在环\(\mathcal {R}\) 上为\((\theta , \delta _\theta )\) -循环码生成多项式和幂等生成器。此外，表明主要生成了\((\theta , \delta _\theta )\) -循环码。最后，通过使用分解方法，我们提供了几个例子\((\theta , \delta _\theta )\) - 不同长度的循环码在\(\mathcal {R}\) 其中许多是根据可用数据库优化的（Grassl，代码表：各种类型代码参数的界限。http://www.codetables.de/）。