Optimal binary codes from ${\mathcal{C}}_D$-codes over a non-chain ring

Abstract

In \cite{shi2022few-weight}, Shi and Li studied ${\mathcal{C}}_D$-codes over the ring $\mathcal{R}:={\mathbb{F}}_2[x,y]/⟨x^2,y^2,xy-yx⟩$ and their binary Gray images, where $D$ is derived using certain simplicial complexes. We study the subfield codes ${\mathcal{C}}_D^{(2)}$ of ${\mathcal{C}}_D$-codes over $\mathcal{R},$ where $D$ is as in \cite{shi2022few-weight} and more. We find the Hamming weight distribution and the parameters of ${\mathcal{C}}_D^{(2)}$ for various $D$, and identify several infinite families of codes that are distance-optimal. Besides, we provide sufficient conditions under which these codes are minimal and self-orthogonal. Two families of strongly regular graphs are obtained as an application of the constructed two-weight codes.