  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.
