Let $\mathscr{R}_{e,m}$ denote a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 3,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathscr{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$ for some odd integer $κ$ with $3 \leq κ\leq e.$ In this paper, we first develop a recursive method to construct a self-orthogonal code $\mathscr{D}_e$ of type $\{λ_1, λ_2, \ldots, λ_e\}$ and length $n$ over $\mathscr{R}_{e,m}$ from a chain $\mathcal{C}^{(1)}\subseteq \mathcal{C}^{(2)} \subseteq \cdots \subseteq \mathcal{C}^{(\lceil \frac{e}{2} \rceil)} $ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, subject to certain conditions, where $λ_1,λ_2,\ldots,λ_e$ 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 $\lceil \frac{e+1}{2} \rceil \leq i\leq e,$ and $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions, respectively. This construction ensures that $Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$ With the help of this recursive construction method and by applying results from group theory and finite geometry, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathscr{R}_{e,m}.$ We also illustrate these results with some examples.

翻译：设 $\mathscr{R}_{e,m}$ 为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 $e \geq 3$，Teichm$\ddot{u}$ller 集为 $\mathcal{T}_{m}$，且剩余域 $\mathscr{R}_{e,m}/\langle u \rangle$ 的阶为 $2^m$。假设对于某个满足 $3 \leq κ\leq e$ 的奇数 $κ$，有 $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$。本文首先发展了一种递归方法，用于从 $\mathcal{T}_{m}$ 上长度为 $n$ 的自正交码链 $\mathcal{C}^{(1)}\subseteq \mathcal{C}^{(2)} \subseteq \cdots \subseteq \mathcal{C}^{(\lceil \frac{e}{2} \rceil)} $（反之亦然）出发，在特定条件下构造 $\mathscr{R}_{e,m}$ 上类型为 $\{λ_1, λ_2, \ldots, λ_e\}$、长度为 $n$ 的自正交码 $\mathscr{D}_e$，其中 $λ_1,λ_2,\ldots,λ_e$ 为非负整数，满足对于 $\lceil \frac{e+1}{2} \rceil \leq i\leq e$，有 $2λ_1+2λ_2+\cdots+2λ_{e-i+1}+λ_{e-i+2}+λ_{e-i+3}+\cdots+λ_i \leq n$，且 $\lfloor \cdot \rfloor$ 和 $\lceil \cdot \rceil$ 分别表示向下取整和向上取整函数。此构造确保了对于 $1 \leq i \leq \lceil \frac{e}{2} \rceil$，有 $Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)}$。借助此递归构造方法，并应用群论和有限几何中的结果，我们得到了 $\mathscr{R}_{e,m}$ 上任意长度的所有自正交码与自对偶码的显式枚举公式。我们还通过若干示例对这些结果进行了说明。