Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic - II
Monika Yadav, Anuradha Sharma
TL;DR
The paper addresses the problem of constructing and enumerating self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic. It introduces a refined recursive approach that lifts chains of self-orthogonal codes over the Teichmüller set $\mathcal{T}_m$ to codes over $\mathcal{R}_{e,m}$, with careful control of torsion codes via $Tor_i(\mathcal{C}_e)$ and two main regimes determined by $2\kappa \le e$ or $2\kappa > e$. The authors establish existence results under explicit A1–A5 and B1–B4 conditions, and derive comprehensive enumeration formulae expressed through Gaussian binomial coefficients and auxiliary combinatorial terms, applicable to arbitrary length and type $\{\lambda_1,\dots,\lambda_e\}$. They also provide concrete examples and computational verification, highlighting the practical utility for code classification and related algebraic structures in areas like quantum error correction and code-based cryptography. Overall, the work provides a complete theoretical framework and practical tools for the construction and counting of these codes over chain rings in the even characteristic setting.
Abstract
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.