Let \({\mathcal {R}}_e=GR(p^e,r)[y]/\langle g(y),p^{e-1}y^t\rangle \) be a finite commutative chain ring, where p is a prime number, the integers e, r, t, k satisfy \(e \ge 2,\) \( r \ge 1\) and \(1\le t\le k,\) \(GR(p^e,r)\) is the Galois ring of characteristic \(p^e\) and rank r, and \(g(y)\in GR(p^e,r)[y]\) is an Eisenstein polynomial of degree k. In this paper, we first establish a duality-preserving 1-1 correspondence between additive codes over \({\mathcal {R}}_e\) and \({\mathbb {Z}}_{p^e}{\mathbb {Z}}_{p^{e-1}}\) -linear codes, where the character-theoretic dual codes of additive codes over \({\mathcal {R}}_e\) correspond to the Euclidean dual codes of \({\mathbb {Z}}_{p^e}{\mathbb {Z}}_{p^{e-1}}\) -linear codes, and vice versa. This correspondence gives rise to a construction method for obtaining additive codes over \({\mathcal {R}}_e\) and their character-theoretic dual codes, since, unlike additive codes over \({\mathcal {R}}_e,\) \({\mathbb {Z}}_{p^e}{\mathbb {Z}}_{p^{e-1}}\) -linear codes can be completely described in terms of generator matrices. We also list additive codes over the chain ring \({\mathbb {Z}}_4[y]/\langle y^2-2,2y \rangle \) achieving the Plotkin’s bound for homogeneous weights, which suggests that additive codes over \({\mathcal {R}}_e\) is a promising class of error-correcting codes to find optimal codes with respect to the homogeneous metric. We further provide a method to construct and enumerate all Euclidean self-orthogonal and self-dual \(\texttt{R}_e\texttt{R}_{e-1}\) -linear codes of an arbitrary block-length, where \(\texttt{R}_e\) is a finite commutative chain ring of odd characteristic with maximal ideal \(\langle \gamma \rangle \) of nilpotency index \(e\ge 2\) and \(\texttt{R}_{e-1} = \texttt{R}_e/\langle \gamma ^{e-1}\rangle \) is the chain ring with maximal ideal of nilpotency index \(e-1.\) By employing this method, we obtain enumeration formulae for all Euclidean self-orthogonal and self-dual \(\texttt{R}_e\texttt{R}_{e-1}\) -linear codes of an arbitrary block-length. This also gives rise to a construction method and enumeration formulae for all self-orthogonal and self-dual additive codes over \({\mathcal {R}}_e,\) where p is an odd prime. We also obtain an enumeration formula for all complementary-dual additive codes (or ACD codes in short) over \({\mathcal {R}}_e.\) Besides this, we translate the concept of monomial equivalence between additive codes over \({\mathcal {R}}_e\) to a suitable notion of equivalence between \({\mathbb {Z}}_{p^e}{\mathbb {Z}}_{p^{e-1}}\) -linear codes. With the help of this observation and our enumeration formulae, we classify all self-orthogonal and self-dual additive codes of lengths 2 and 3 over the chain ring \({\mathbb {Z}}_9[y]/\langle y^2-3,3y\rangle \) up to monomial equivalence by classifying all Euclidean self-orthogonal and self-dual \({\mathbb {Z}}_9{\mathbb {Z}}_3\) -linear codes of block-lengths (2, 2) and (3, 3), respectively. We also classify all ACD codes of length 2 over \({\mathbb {Z}}_4[y]/\langle y^2-2,2y\rangle \) and \({\mathbb {Z}}_9[y]/\langle y^2-3,3y\rangle \) up to monomial equivalence by classifying all Euclidean complementary-dual \({\mathbb {Z}}_4{\mathbb {Z}}_2\) -linear codes (or Euclidean \({\mathbb {Z}}_4{\mathbb {Z}}_2\) -LCD codes in short) and all Euclidean complementary-dual \({\mathbb {Z}}_9{\mathbb {Z}}_3\) -linear codes (or Euclidean \({\mathbb {Z}}_9{\mathbb {Z}}_3\) -LCD codes in short) of block-length (2, 2), respectively.
