Let $\mathcal{R}_{e,m}$ be a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 2,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathcal{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$ for some even positive integer $ κ\leq e.$ In this paper, we provide a recursive method to construct a self-orthogonal code $\mathcal{C}_e$ of type $\{λ_1, λ_2, \ldots, λ_e\}$ and length $n$ over $\mathcal{R}_{e,m}$ from a chain $\mathcal{D}^{(1)}\subseteq \mathcal{D}^{(2)} \subseteq \cdots \subseteq \mathcal{D}^{(\lceil \frac{e}{2} \rceil)}$ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, where $\dim \mathcal{D}^{(i)}=λ_1+λ_2+\cdots+λ_i$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil,$ the codes $\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor-κ)},\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor -κ+1)},\ldots,\mathcal{D}^{(\lfloor \frac{e}{2}\rfloor-\lfloor \fracκ{2} \rfloor)}$ satisfy certain additional conditions, and $λ_1,λ_2,\ldots,λ_e$ are non-negative integers satisfying $2λ_1+2λ_2+\cdots+2λ_{e-i+1}+λ_{e-i+2}+λ_{e-i+3}+\cdots+λ_i \leq n$ for $\lceil \frac{e+1}{2} \rceil \leq i\leq e.$ This construction guarantees that $Tor_i(\mathcal{C}_e)=\mathcal{D}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$ By employing this recursive construction method, together with the results from group theory and finite geometry, we derive explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathcal{R}_{e,m}.$ We also demonstrate these results through examples.

翻译：设 $\mathcal{R}_{e,m}$ 是一个特征为偶数的有限交换链环，其极大理想为 $\langle u \rangle$，幂零指数为 $e \geq 2$，Teichm$\ddot{u}$ller 集为 $\mathcal{T}_{m}$，且剩余域 $\mathcal{R}_{e,m}/\langle u \rangle$ 的阶为 $2^m$。假设对于某个偶数正整数 $κ \leq e$，有 $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$。本文提出了一种递归方法，用于从 $\mathcal{T}_{m}$ 上长度为 $n$ 的自正交码链 $\mathcal{D}^{(1)}\subseteq \mathcal{D}^{(2)} \subseteq \cdots \subseteq \mathcal{D}^{(\lceil \frac{e}{2} \rceil)}$ 构造 $\mathcal{R}_{e,m}$ 上类型为 $\{λ_1, λ_2, \ldots, λ_e\}$、长度为 $n$ 的自正交码 $\mathcal{C}_e$，反之亦然。其中，对于 $1 \leq i \leq \lceil \frac{e}{2} \rceil$，有 $\dim \mathcal{D}^{(i)}=λ_1+λ_2+\cdots+λ_i$；码 $\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor-κ)},\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor -κ+1)},\ldots,\mathcal{D}^{(\lfloor \frac{e}{2}\rfloor-\lfloor \fracκ{2} \rfloor)}$ 满足某些附加条件；且 $λ_1,λ_2,\ldots,λ_e$ 是非负整数，满足对于 $\lceil \frac{e+1}{2} \rceil \leq i\leq e$，有 $2λ_1+2λ_2+\cdots+2λ_{e-i+1}+λ_{e-i+2}+λ_{e-i+3}+\cdots+λ_i \leq n$。此构造保证了对于 $1 \leq i \leq \lceil \frac{e}{2} \rceil$，有 $Tor_i(\mathcal{C}_e)=\mathcal{D}^{(i)}$。通过运用此递归构造方法，并结合群论与有限几何的结果，我们推导出了 $\mathcal{R}_{e,m}$ 上任意长度的所有自正交码与自对偶码的显式枚举公式。我们还通过示例展示了这些结果。