标题： 递归构造和枚举有限交换链环的偶数特征的自正交和自对偶码 - 二 显示英文标题

标题： Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic - II

摘要： 设$\mathcal{R}_{e,m}$是一个偶数特征的有限交换链环，其极大理想$\langle u \rangle$的幂零指数为$e \geq 2,$，Teichm$\ddot{u}$ller 集$\mathcal{T}_{m},$和剩余域$\mathcal{R}_{e,m}/\langle u \rangle$，其阶为$2^m.$。假设对于某个正偶数$ \kappa \leq e.$，$2 \in \langle u^{\kappa}\rangle \setminus \langle u^{\kappa+1}\rangle$。在本文中，我们提供了一种递归方法，从长度为$n$的自正交码链$\mathcal{D}^{(1)}\subseteq \mathcal{D}^{(2)} \subseteq \cdots \subseteq \mathcal{D}^{(\lceil \frac{e}{2} \rceil)}$在$\mathcal{T}_{m},$上，反之亦然，构造类型为$\{\lambda_1, \lambda_2, \ldots, \lambda_e\}$且长度为$n$的自正交码$\mathcal{C}_e$，在$\mathcal{R}_{e,m}$上，其中$\dim \mathcal{D}^{(i)}=\lambda_1+\lambda_2+\cdots+\lambda_i$对于$1 \leq i \leq \lceil \frac{e}{2} \rceil,$，码$\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor-\kappa)},\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor -\kappa+1)},\ldots,\mathcal{D}^{(\lfloor \frac{e}{2}\rfloor-\lfloor \frac{\kappa}{2} \rfloor)}$满足某些附加条件，且$\lambda_1,\lambda_2,\ldots,\lambda_e$是满足$2\lambda_1+2\lambda_2+\cdots+2\lambda_{e-i+1}+\lambda_{e-i+2}+\lambda_{e-i+3}+\cdots+\lambda_i \leq n$对于$\lceil \frac{e+1}{2} \rceil \leq i\leq e.$的非负整数。此构造保证了$Tor_i(\mathcal{C}_e)=\mathcal{D}^{(i)}$对于$1 \leq i \leq \lceil \frac{e}{2} \rceil.$。通过采用这种递归构造方法，并结合群论和有限几何的结果，我们得出了任意长度的自正交码和自对偶码在$\mathcal{R}_{e,m}.$上的显式计数公式。我们还通过例子展示了这些结果。 显示英文摘要

摘要： 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 $2 \in \langle u^{\kappa}\rangle \setminus \langle u^{\kappa+1}\rangle$ for some even positive integer $ \kappa \leq e.$ In this paper, we provide a recursive method to construct a self-orthogonal code $\mathcal{C}_e$ of type $\{\lambda_1, \lambda_2, \ldots, \lambda_e\}$ 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 $\dim \mathcal{D}^{(i)}=\lambda_1+\lambda_2+\cdots+\lambda_i$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil,$ the codes $\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor-\kappa)},\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor -\kappa+1)},\ldots,\mathcal{D}^{(\lfloor \frac{e}{2}\rfloor-\lfloor \frac{\kappa}{2} \rfloor)}$ satisfy certain additional conditions, and $\lambda_1,\lambda_2,\ldots,\lambda_e$ are non-negative integers satisfying $2\lambda_1+2\lambda_2+\cdots+2\lambda_{e-i+1}+\lambda_{e-i+2}+\lambda_{e-i+3}+\cdots+\lambda_i \leq n$ 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.