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

Abstract

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 ${\mathcal{R}}_2:={\mathbb{F}}_2[x]/⟨x^3-x⟩$ 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 $Δ_X:=\{v\in \mathbb{F}_2^m: \Supp(v)\subseteq X\}$ and $D:=(1+u^2)D_1+u^2{D}_2+(u+u^2)D_3,$ a subset of ${\mathcal{R}}_2^m,$ where $u=x+⟨x^3-x⟩,D_1\in \{{\Delta }_L,{\Delta }_L^c\},D_2\in \{{\Delta }_M,{\Delta }_M^c\}$ 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.