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, $GR(p^e,r)$ is the Galois ring of characteristic $p^e$ and rank $r,$ $t$ and $k$ are positive integers satisfying $1\leq t\leq k$ when $e \geq 2,$ while $t=k$ when $e=1,$ and $g(y)=y^k+p(g_{k-1}y^{k-1}+\cdots+g_1y+g_0)\in GR(p^e,r)[y]$ is an Eisenstein polynomial with $g_0$ as a unit in $GR(p^e,r).$ 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 method for constructing additive codes over $\mathcal{R}_e$ and their character-theoretic dual codes, as 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.

翻译：令 $\mathcal{R}_e=GR(p^e,r)[y]/\langle g(y),p^{e-1}y^t\rangle$ 为一个有限交换链环，其中 $p$ 为素数，$GR(p^e,r)$ 是特征为 $p^e$、秩为 $r$ 的伽罗瓦环，$t$ 和 $k$ 是满足 $1\leq t\leq k$（当 $e \geq 2$ 时）或 $t=k$（当 $e=1$ 时）的正整数，且 $g(y)=y^k+p(g_{k-1}y^{k-1}+\cdots+g_1y+g_0)\in GR(p^e,r)[y]$ 是一个以 $g_0$ 作为 $GR(p^e,r)$ 中单位的艾森斯坦多项式。本文首先建立了 $\mathcal{R}_e$ 上的加性码与 $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-线性码之间保持对偶性的一一对应关系，其中 $\mathcal{R}_e$ 上加性码的特征理论对偶码对应于 $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-线性码的欧几里得对偶码，反之亦然。这一对应关系提供了一种构造 $\mathcal{R}_e$ 上加性码及其特征理论对偶码的方法，因为与 $\mathcal{R}_e$ 上的加性码不同，$\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-线性码可以完全通过生成矩阵来描述。我们还列举了在链环 $\mathbb{Z}_4[y]/\langle y^2-2,2y \rangle$ 上达到齐次权重普洛特金界的加性码，这表明 $\mathcal{R}_e$ 上的加性码是一类有望在齐次度量下找到最优码的纠错码。