In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

翻译：本文研究有限环$\mathcal{S}$上线性码的构造，其中$\mathcal{S}$取为阶为$p^2$的交换非幺环$I$的扩环。我们的方法基于定义集方法。本文考虑的定义集来源于可能包含多个极大元的一般单纯复形。我们确定了这些码在$\mathcal{S}$上的参数，并研究了它们的Gray像。我们还研究了相应的子域类码。我们证明，这些Gray像码和子域类码能产生若干族可分码。此外，我们建立了这些码为极小码、最优码和自正交码的充分条件。作为我们结果的应用，我们获得了若干族投影少重码和具有小局部性的局部可恢复码。我们还研究了与这些极小码的对偶码相关的秘密共享方案的最小访问结构。此外，我们从投影二重码构造了若干族强正则图，并明确确定了它们的参数。