Subfield codes of \(C_D\)-codes over \(\mathbb{F}_2[x]/\langle x^3-x \rangle\) 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 \(\mathcal{R}_2:=\mathbb{F}_2[x]/\langle x^3-x\rangle\) 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.