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

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 ${\mathcal{R}}_{e,m}\mathrm{/}⟨u⟩$ of order $2^m\mathrm{.}$ Suppose that $2\in ⟨u^{\kappa }⟩\setminus ⟨u^{\kappa +1}⟩$ for some odd integer $\kappa$ with $3\le \kappa \le e\mathrm{.}$ In this paper, we first develop a recursive method to construct a self-orthogonal code ${\mathcal{D}}_e$ of type $\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_e\}$ 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 ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_e$ are non-negative integers satisfying $2{\lambda }_1+2{\lambda }_2+\cdots +2{\lambda }_{e-i+1}+{\lambda }_{e-i+2}+{\lambda }_{e-i+3}+\cdots +{\lambda }_i\le n$ 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 $1\le i\le \lceil \frac{e}{2}\rceil \mathrm{.}$ 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 ${\mathcal{R}}_{e,m}\mathrm{.}$ We also illustrate these results with some examples.