In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee weight distributions. As a result, we find at least 32 new or improved quaternary linear codes as per the database \cite{aydin2022updated} of best-known quaternary codes, including codes from a Plotkin-optimal family. We also report 6 projective quaternary linear codes with best-known parameters that might outperform the currently reported best-known codes due to their projectivity. Further, we establish necessary and sufficient conditions for their Gray image to be linear, which in turn gives an infinite family of Griesmer codes and several infinite families of minimal binary linear codes.

翻译：本文构造了无穷族四元（即环$\mathbb{Z}_4$上的）$\mathcal{C}_{D}$-码，其中定义集$D$利用二生成单纯复形导出，并确定了它们的Lee重量分布。由此，我们在已知最优四元码数据库\cite{aydin2022updated}中找到了至少32个新的或改进的四元线性码，包括来自Plotkin最优族的码。我们还报告了6个具有已知最优参数的射影四元线性码，这些码由于其射影性可能优于当前已报告的最优码。进一步地，我们建立了其Gray像为线性的充要条件，这进而给出了一个无穷Griesmer码族和多个无穷极小二元线性码族。