Subfield codes of $C_D$-codes over $\mathbb{F}_2[x]/\langle x^3-x \rangle$ are really nice!

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 $\DeltaX:={v\in \mathbb{F}2^m: \Supp(v)\subseteq X}$ and $D:=(1+u^{2)D1+u2D2+(u+u2)D_3,$} a subset of $\mathcal{R}2^m,$ where $u=x+\langle x^3-x\rangle, D1\in {\DeltaL, \DeltaL^c},\, D2\in {\DeltaM, \DeltaM^c}$ and $ D3\in {\DeltaN, \DeltaN^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.