Certain binary minimal codes constructed using simplicial complexes

In this article, we work over the non-chain ring $\mathcal{R} = \mathbb{Z}_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 $\Delta_{X} := \{v \in \mathbb{Z}_2^m : \textnormal{Supp}(v)\subseteq X\}$ and $D:= (1+u^2)D_1 + u^2\big(D_2 + (u+u^2)D_3\big)$, a subset of $\mathcal{R}^m$, 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 $D$. For instance, we obtain the Lee weight distribution of $C_D$. The Gray map $\Phi: \mathcal{R} \longrightarrow \mathbb{Z}_2^3 $ given by $\Phi(a+ub+u^2d) = (a+b, b+d, d)$ is utilized to derive a binary linear code, namely, $\Phi(C_D)$ for each $D$. Sufficient conditions for each of these binary linear codes to be minimal are obtained. In fact, sufficient conditions for minimality are mild in nature, for example, $\vert L\vert , \vert M\vert , \vert N\vert < m-2 $ is a set of conditions for minimality of $\Phi(C_D)$ for each $D$. Moreover, these binary codes are self-orthogonal if each of $L, M$ and $N$ is nonempty.