Let $R={\mathbb{F}}_2+u{\mathbb{F}}_2$ $(u^2=0)$ and $S={\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$ $(u^3=0)$ be two finite commutative chain rings. This paper studies ${\mathbb{F}}_{2}RS$-cyclic codes, which are described as $S\left[x\right]$-submodules of the $S\left[x\right]$-module ${\mathbb{F}}_2\left[x\right]∕〈x^r-1〉\times R\left[x\right]∕〈x^s-1〉\times S\left[x\right]∕〈x^t-1〉$. We study their generator polynomials and the minimal generating sets. We classify each case of the generating sets separately and determine the size of each such case. Free ${\mathbb{F}}_{2}RS$-cyclic codes and separable codes are discussed, and the structural properties of dual codes of free ${\mathbb{F}}_{2}RS$-cyclic codes are investigated. Moreover, we determine the relationship between the generator polynomials of free ${\mathbb{F}}_{2}RS$-cyclic codes and their duals. As applications, we provide several examples of optimal and near-optimal binary codes which are obtained from the Gray images of ${\mathbb{F}}_{2}RS$-cyclic codes.

让 $\mathrm{[R}={\mathbb{F}}_2+ü{\mathbb{F}}_2$ $（{ü}^2=0）$ 和 $\mathrm{小号}={\mathbb{F}}_2+ü{\mathbb{F}}_2+{ü}^2{\mathbb{F}}_2$ $（{ü}^3=0）$是两个有限的可交换链环。本文研究${\mathbb{F}}_2\mathrm{[R}\mathrm{小号}$-循环码，描述为 $\mathrm{小号}\left[X\right]$-的子模块 $\mathrm{小号}\left[X\right]$-模块 ${\mathbb{F}}_2\left[X\right]∕〈X^{\mathrm{[R}}-\mathrm{1个}〉\times \mathrm{[R}\left[X\right]∕〈X^s-\mathrm{1个}〉\times \mathrm{小号}\left[X\right]∕〈X^{Ť}-\mathrm{1个}〉$。我们研究了它们的生成多项式和最小生成集。我们将发电机组的每种情况分别分类，并确定每种情况的大小。自由${\mathbb{F}}_2\mathrm{[R}\mathrm{小号}$讨论了循环码和可分离码，以及自由对偶码的结构性质 ${\mathbb{F}}_2\mathrm{[R}\mathrm{小号}$-循环码被研究。此外，我们确定自由的生成多项式之间的关系${\mathbb{F}}_2\mathrm{[R}\mathrm{小号}$-循环码及其对偶。作为应用程序，我们提供了几个最佳和接近最佳的二进制代码的示例，这些示例是从${\mathbb{F}}_2\mathrm{[R}\mathrm{小号}$-循环代码。