In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over $\mathbb{F}_q$ are also obtained, where $q$ is a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case of $q=2,3,4$. It is noteworthy that most of the derived codes over $\mathbb{F}_q$ satisfy the Ashikhmin-Barg's condition for minimality. Besides, we obtain two infinite families of distance-optimal codes over $\mathbb{F}_q$ with respect to the Griesmer bound.

翻译：本文利用一般单纯复形在非幺非交换环的扩环上构造了若干无限族码。得益于定义集的特珠结构，这些码的主要参数得以刻画。特别地，当所采用的单纯复形由单个极大元生成时，我们完全确定了其Lee重量分布。此外，通过考虑Gray像码及相应的类子域码，我们还得到了$\mathbb{F}_q$上的大量线性码，其中$q$为素数幂。在$q=2,3,4$的情形下，给出了确保上述线性码为（Hermitian）自正交的若干条件。值得注意的是，所导出的$\mathbb{F}_q$上绝大多数码满足Ashikhmin-Barg极小性条件。此外，我们得到了关于Griesmer界的两个无限族距离最优$\mathbb{F}_q$码。