A recursive approach to the construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic

Abstract

Let ${\mathcal{R}}_{e,m}$ be a finite commutative chain ring of even characteristic with maximal ideal $⟨u⟩$ of nilpotency index $e\ge 2,$ Teichm$\ddot{u}$ller set ${\mathcal{T}}_m,$ and residue field ${\mathcal{R}}_{e,m}/⟨u⟩$ of order $2^m.$ Suppose that $2\in ⟨u^{\kappa }⟩\setminus ⟨u^{\kappa +1}⟩$ for some even positive integer $\kappa \le e.$ In this paper, we provide a recursive method to construct a self-orthogonal code ${\mathcal{C}}_e$ of type $\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_e\}$ 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 $1\le i\le \lceil \frac{e}{2}\rceil ,$ the codes ${\mathcal{D}}^{(\lfloor \frac{e+1}{2}\rfloor -\kappa )},{\mathcal{D}}^{(\lfloor \frac{e+1}{2}\rfloor -\kappa +1)},\dots ,{\mathcal{D}}^{(\lfloor \frac{e}{2}\rfloor -\lfloor \frac{\kappa }{2}\rfloor )}$ satisfy certain additional conditions, and ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_e$ 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 $\lceil \frac{e+1}{2}\rceil \le i\le e.$ This construction guarantees that $To{r}_i({\mathcal{C}}_e)={\mathcal{D}}^{(i)}$ for $1\le i\le \lceil \frac{e}{2}\rceil .$ 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.