Structure and Construction of Two-Dimensional Minimal Linear Codes over the rings $\mathbb{Z}_{p^n}$ with Applications to Secret Sharing

Abstract

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.