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

Abstract

Let ${\mathcal{R}}_{e,m}$ denote a finite commutative chain ring of even characteristic with maximal ideal $⟨u⟩$ of nilpotency index $e\ge 3,$ Teichm$\ddot{u}$ller set ${\mathcal{T}}_m,$ and residue field ${\mathcal{R}}_{e,m}/⟨u⟩$ of order $2^m.$ Suppose that $2\in ⟨u^{\hat{I}}º⟩\setminus ⟨u^{\hat{I}º+1}⟩$ for some odd integer $\hat{I}º$ with $3\le \hat{I}º\le e.$ In this paper, we first develop a recursive method to construct a self-orthogonal code ${\mathcal{D}}_e$ of type $\{\hat{I}{»}_1,\hat{I}{»}_2,\dots ,\hat{I}{»}_e\}$ and length $n$ over ${\mathcal{R}}_{e,m}$ 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 $\hat{I}{»}_1,\hat{I}{»}_2,\dots ,\hat{I}{»}_e$ are non-negative integers satisfying $2\hat{I}{»}_1+2\hat{I}{»}_2+\cdots +2\hat{I}{»}_{e-i+1}+\hat{I}{»}_{e-i+2}+\hat{I}{»}_{e-i+3}+\cdots +\hat{I}{»}_i\le n$ for $\lceil \frac{e+1}{2}\rceil \le i\le e,$ and $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions, respectively. This construction ensures that $To{r}_i({\mathcal{D}}_e)={\mathcal{C}}^{(i)}$ for $1\le i\le \lceil \frac{e}{2}\rceil .$ 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 ${\mathcal{R}}_{e,m}.$ We also illustrate these results with some examples.

arXiv admin note: substantial text overlap with arXiv:2510.06069