Let \({\mathscr {R}}_{e,m}\) denote a finite commutative chain ring of even characteristic with maximal ideal \(\langle u \rangle \) of nilpotency index \(e \ge 3,\) Teichm \(\ddot{u}\) ller set \({\mathcal {T}}_{m},\) and residue field \({\mathscr {R}}_{e,m}/\langle u \rangle \) of order \(2^m.\) Suppose that \(2 \in \langle u^{\kappa }\rangle \setminus \langle u^{\kappa +1}\rangle \) for some odd integer \(\kappa \) with \(3 \le \kappa \le e.\) In this paper, we first develop a recursive method to construct a self-orthogonal code \({\mathscr {D}}_e\) of type \(\{\lambda _1, \lambda _2, \ldots , \lambda _e\}\) 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 the following conditions: (i) \(\dim {\mathcal {C}}^{(i)}=\lambda _1+\lambda _2+\cdots +\lambda _i\) for \(1 \le i \le \lceil \frac{e}{2}\rceil ;\)
(ii) the code \({\mathcal {C}}^{({\lfloor \frac{e}{2}\rfloor }-{\lfloor \frac{\kappa }{2}\rfloor })}\) is doubly even; and
(iii) the all-one vector \({\textbf {1}} =(1,1,\ldots ,1)\notin {\mathcal {C}}^{(\lceil \frac{e}{2}\rceil -\kappa )}\) provided that \(2\kappa \le e,\) \(n\equiv 4\pmod 8\) and m is odd,
where \(\lambda _1,\lambda _2,\ldots ,\lambda _e\) 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 \(Tor_i({\mathscr {D}}_e)={\mathcal {C}}^{(i)}\) 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 \({\mathscr {R}}_{e,m}.\) We also illustrate these results with some examples. In a subsequent study [33], we will address the complementary case where \(\kappa \) is even.
