Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic - I
Monika Yadav, Anuradha Sharma
TL;DR
This work develops a modified recursive framework to construct and enumerate self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic, focusing on the case where $p=2$ and $\kappa$ is odd with $3\le\kappa\le e$. by lifting a chain of self-orthogonal codes over the Teichmüller set $\mathcal{T}_m$ to codes over $\mathscr{R}_{e,m}$, the authors provide existence results and exact counts for all codes of a given type, under detailed parity and doubly-even constraints, using constructions (A) and (B) and a property $(\mathfrak{P})$ that governs the lifting across levels. They define torsion codes, derive dimension relations $\dim\,\mathrm{Tor}_i(\mathscr{D})$, and establish self-orthogonality criteria via matrix congruences, enabling recursive lifting to $\mathscr{R}_{e,m}$ with $Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)}$ for $i\le\lceil e/2\rceil$. The paper then delivers explicit enumeration formulas for all self-orthogonal and self-dual codes of length $n$ over $\mathscr{R}_{e,m}$, expressed through Gaussian binomial coefficients over $2^m$ and counts of doubly-even codes over the Teichmüller set, with separate treatments for odd/even $e$, parity of $n$, and related parameters. These results advance the classification and counting of codes over even-characteristic chain rings and lay groundwork for extensions to even-$\kappa$ and broader Frobenius ring settings, with applications to quantum codes and combinatorial designs.
Abstract
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.