Subfield codes of CD-codes over $F_2[x]/ x^3-x $ are really nice!

A non-zero F-linear map from a finite-dimensional commutative F-algebra to F is called an F-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an F₂-valued trace of the F₂-algebra R₂:=F₂[x]/ x³-x to study binary subfield code CD⁽²⁾ of CD:=\(x· d)d∈ D: x∈ R₂ᵐ\ for each defining set D derived from a certain simplicial complex. For m∈ N and X⊆ \1, 2, , m\, define ΔX:=∈ F₂ᵐ: (v)⊆ X\ and D:=(1+u²)D₁+u²D₂+(u+u²)D₃, a subset of R₂ᵐ, where u=x+ x³-x, D₁∈ \ΔL, ΔLᶜ\,\, D₂∈ \ΔM, ΔMᶜ\ and D₃∈ \ΔN, ΔNᶜ\, for L, M, N⊆ \1, 2, , m\. The parameters and the Hamming weight distribution of the binary subfield code CD⁽²⁾ of CD 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.