Subfield codes of CD-codes over F₂x/ x³-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: =F₂x/ x³-x to study binary subfield code CD^ (2) of CD: =\ (x d) ₃ ₃: x R₂ᵐ\ for each defining set D derived from a certain simplicial complex. For m N and X \1, 2, , m\, define X: =\v F₂ᵐ: (v) X\ and D: = (1+u²) D₁+u²D₂+ (u+u²) D₃, a subset of R₂ᵐ, where u=x+ x³-x, D₁ \L, Lᶜ\, \, D₂ \M, Mᶜ\ and D₃ \N, Nᶜ\, 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.