Let ${\mathbb{F}}_{2^m}$ be the finite field of $2^m$ elements and $s$ be any positive integer. The existing literature only gives an effective calculation method to represent all distinct Euclidean self-dual cyclic codes of length $2^s$ over the finite chain ring ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ $(u^2=0)$, such as in Cao et al., (2019). As a development of this topic, we provide an explicit expression for each of these self-dual cyclic codes, using binomial coefficients. The Gray image of any self-dual cyclic code over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ of length $2^s$ is a self-dual $2$-quasi-cyclic code over ${\mathbb{F}}_{2^m}$ of length $2^{s+1}$. In particular, we give a generator matrix for each of these self-dual $2$-quasi-cyclic codes over ${\mathbb{F}}_{2^m}$.

让 ${\mathbb{F}}_{2^{米}}$ 是...的有限域 $2^{米}$ 元素和 $s$是任何正整数。现有文献仅给出了一种有效的计算方法来表示所有长度不同的欧几里德自对偶循环码$2^s$ 在有限链环上 ${\mathbb{F}}_{2^{米}}+ü{\mathbb{F}}_{2^{米}}$ $（{ü}^2=0）$（例如Cao等，（2019））。作为该主题的发展，我们使用二项式系数为这些自对偶循环码提供了一个明确的表达式。任何自对偶循环码的灰度图像${\mathbb{F}}_{2^{米}}+ü{\mathbb{F}}_{2^{米}}$ 长度 $2^s$ 是自我对偶 $2$-准循环编码 ${\mathbb{F}}_{2^{米}}$ 长度 $2^{s+\mathrm{1个}}$。特别是，我们为每个自对偶给出一个生成器矩阵$2$-准循环码 ${\mathbb{F}}_{2^{米}}$。