In this manuscript, we work over the non-chain ring $\mathcal{R} = \mathbb{F}_2[u]/\langle u^3 - u\rangle $. Let $m\in \mathbb{N}$ and let $L, M, N \subseteq [m]:=\{1, 2, \dots, m\}$. For $X\subseteq [m]$, define $\Delta_X:=\{v \in \mathbb{F}_2^m : \textnormal{Supp}(v)\subseteq X\}$ and $D:= (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3$, an ordered finite multiset consisting of elements from $\mathcal{R}^m$, where $D_1\in \{\Delta_L, \Delta_L^c\}, D_2\in \{\Delta_M, \Delta_M^c\}, D_3\in \{\Delta_N, \Delta_N^c\}$. The linear code $C_D$ over $\mathcal{R}$ defined by $\{\big(v\cdot d\big)_{d\in D} : v \in \mathcal{R}^m \}$ is studied for each $D$. Further, we also consider simplicial complexes with two maximal elements in the above work. We study their binary Gray images and the binary subfield-like codes corresponding to a certain $\mathbb{F}_{2}$-functional of $\mathcal{R}$. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.

翻译：本文在非链环 $\mathcal{R} = \mathbb{F}_2[u]/\langle u^3 - u\rangle$ 上展开研究。设 $m\in \mathbb{N}$ 且 $L, M, N \subseteq [m]:=\{1, 2, \dots, m\}$。对于 $X\subseteq [m]$，定义 $\Delta_X:=\{v \in \mathbb{F}_2^m : \textnormal{Supp}(v)\subseteq X\}$ 以及 $D:= (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3$，这是一个由 $\mathcal{R}^m$ 中元素构成的有序有限多重集，其中 $D_1\in \{\Delta_L, \Delta_L^c\}, D_2\in \{\Delta_M, \Delta_M^c\}, D_3\in \{\Delta_N, \Delta_N^c\}$。我们针对每个 $D$ 研究了由 $\{\big(v\cdot d\big)_{d\in D} : v \in \mathcal{R}^m \}$ 定义的 $\mathcal{R}$ 上的线性码 $C_D$。此外，在上述工作中我们还考虑了具有两个极大元的单纯复形。我们研究了这些码的二元Gray像以及对应于 $\mathcal{R}$ 的某个 $\mathbb{F}_{2}$-泛函的二元子域类码。在每种情形下，我们得到了这些二元线性码为极小码和自正交码的充分条件。同时，我们构造了一个关于Griesmer界的最优码无穷族。本文得到的大部分码都是少重量码。