Minimal linear codes play an important role in coding theory and cryptography, particularly in the construction of secret sharing schemes. In this paper, we investigate the structure and construction of two-dimensional minimal linear codes over the finite rings $\mathbb{Z}_{p^n}$. We provide an explicit construction of a family of two-dimensional linear codes generated by a structured $2\times m$ matrix over $\mathbb{Z}_{p^n}$ and prove that these codes are minimal whenever the generator matrix contains all $p^n+p^{n-1}$ essential types of column vectors. We further show that this condition is necessary: removing any of these column types destroys the resulting code's minimality. As a consequence, we establish a lower bound on the length of two-dimensional minimal linear codes over $\mathbb{Z}_{p^n}$. Several examples are presented to illustrate the construction and to verify the theoretical results. We also demonstrate that the proposed construction cannot be extended in a straightforward manner to rings of the form $\mathbb{Z}_{p^n q^l}$. Finally, we apply our results to the design of secret sharing schemes derived from minimal linear codes over $\mathbb{Z}_{p^n}$ and analyze the corresponding access structures. Our study highlights structural differences between minimal codes defined over finite rings and those over finite fields, revealing new perspectives for coding-theoretic constructions in cryptographic applications.

翻译：最小线性码在编码理论与密码学中扮演着重要角色，尤其是在秘密共享方案的构造中。本文研究了有限环$\mathbb{Z}_{p^n}$上二维最小线性码的结构与构造。我们给出了一类由$\mathbb{Z}_{p^n}$上结构化$2\times m$矩阵生成的二维线性码的显式构造，并证明了当生成矩阵包含全部$p^n+p^{n-1}$种本质列向量类型时，这些码是最小码。我们进一步证明该条件是必要的：移除任何一种列类型都将破坏所得码的最小性。由此，我们建立了$\mathbb{Z}_{p^n}$上二维最小线性码长度的下界。通过若干例子说明构造过程并验证理论结果。我们还证明所提出的构造无法直接推广到形如$\mathbb{Z}_{p^n q^l}$的环。最后，将结果应用于基于$\mathbb{Z}_{p^n}$上最小线性码导出的秘密共享方案设计，并分析了相应的访问结构。本研究揭示了有限环上最小码与有限域上最小码之间的结构差异，为编码理论在密码学应用中的构造提供了新视角。