  A non-zero $\mathbb{F}$-linear map from a finite-dimensional commutative
$\mathbb{F}$-algebra to $\mathbb{F}$ is called an $\mathbb{F}$-valued trace if
its kernel does not contain any non-zero ideals. In this article, we utilize an
$\mathbb{F}_2$-valued trace of the $\mathbb{F}_2$-algebra
$\mathcal{R}_2:=\mathbb{F}_2[x]/\langle x^3-x\rangle$ to study binary subfield
code $\mathcal{C}_D^{(2)}$ of $\mathcal{C}_D:=\{\left(x\cdot d\right)_{d\in D}:
x\in \mathcal{R}_2^m\}$ for each defining set $D$ derived from a certain
simplicial complex. For $m\in \mathbb{N}$ and $X\subseteq \{1, 2, \dots, m\}$,
define $\Delta_X:=\{v\in \mathbb{F}_2^m: \Supp(v)\subseteq X\}$ and
$D:=(1+u^2)D_1+u^2D_2+(u+u^2)D_3,$ a subset of $\mathcal{R}_2^m,$ where
$u=x+\langle x^3-x\rangle, D_1\in \{\Delta_L, \Delta_L^c\},\, D_2\in
\{\Delta_M, \Delta_M^c\}$ and $ D_3\in \{\Delta_N, \Delta_N^c\}$, for $L, M,
N\subseteq \{1, 2, \dots, m\}.$ The parameters and the Hamming weight
distribution of the binary subfield code $\mathcal{C}_D^{(2)}$ of
$\mathcal{C}_D$ are determined for each $D.$ These binary subfield codes are
minimal under certain mild conditions on the cardinalities of $L, M$ and $N$.
Moreover, most of these codes are distance-optimal. Consequently, we obtain a
few infinite families of minimal, self-orthogonal and distance-optimal binary
linear codes that are either $2$-weight or $4$-weight. It is worth mentioning
that we have obtained several new distance-optimal binary linear codes.
