In this paper, we extend a necessary and sufficient condition for a linear code over a Galois ring to be minimal and establish new bounds on the length of an $m$-dimensional minimal linear code. Building upon this structural characterization, we further generalize the function-based minimality criteria introduced by Wu \emph{et al.} (Cryptogr. Commun. 14, 875-895, 2022) from the finite field setting to the framework of Galois rings. The transition from fields to rings introduces substantial algebraic challenges due to the presence of zero divisors and the richer module structure of $\mathrm{GR}(p^{n},\ell)$. By exploiting Frobenius duality and the chain structure of Galois rings, we derive refined necessary and sufficient conditions ensuring that linear codes arising from functions over $\mathrm{GR}(p^{n},\ell)$ are minimal. As an application of these criteria, we construct several infinite families of minimal linear codes over Galois rings, thereby significantly generalizing the constructions of Wu \emph{et al.} to the ring setting. Our results provide a unified framework that connects minimality theory, module duality over Frobenius rings, and function-based code constructions.

翻译：本文将伽罗华环上线性码为极小码的充要条件进行推广，并建立了 $m$ 维极小线性码长度的新界值。基于这一结构刻画，我们进一步将 Wu 等人 (Cryptogr. Commun. 14, 875-895, 2022) 提出的函数式极小性判据从有限域框架推广至伽罗华环框架。由于零因子的存在以及 $\mathrm{GR}(p^{n},\ell)$ 更丰富的模结构，从域到环的过渡带来了显著的代数挑战。通过利用弗罗贝尼乌斯对偶性与伽罗华环的链结构，我们推导出确保 $\mathrm{GR}(p^{n},\ell)$ 上函数诱导的线性码为极小码的精炼充要条件。作为这些判据的应用，我们构造了多个伽罗华环上极小线性码的无穷族，从而将 Wu 等人的构造在环框架下进行了本质性推广。我们的结果建立了连接极小性理论、弗罗贝尼乌斯环上的模对偶性以及函数式码构造的统一框架。