Subfield codes of CD-codes over F2[x]/〈x3−x〉

Abstract

A non-zero $F$-linear map from a finite-dimensional commutative $F$-algebra to the field $F$ is called an $F$-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an $F_2$-valued trace of the $F_2$-algebra $R_2:=F_2\left[x\right]/〈x^3-x〉$ to study binary subfield code $C_D^{\left(2\right)}$ of $C_D:=\{{(x\cdot d)}_{d\in D}:x\in R_2^m\}$ for each defining set D derived from a certain simplicial complex. For $m\in N$ and $X\subseteq \{1,2,\dots ,m\}$, define ${\Delta }_X:=\{v\in F_2^m:\text{Supp}(v)\subseteq X\}$ and $D:=(1+u^2)D_1+u^2{D}_2+(u+u^2)D_3$, a subset of $R_2^m$, 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 $C_D^{\left(2\right)}$ of $C_D$ 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.