Recently, simplicial complexes are used in constructions of several infinite families of minimal and optimal linear codes by Hyun {\em et al.} Building upon their research, in this paper more linear codes over the ring $\mathbb{Z}_4$ are constructed by simplicial complexes. Specifically, the Lee weight distributions of the resulting quaternary codes are determined and two infinite families of four-Lee-weight quaternary codes are obtained. Compared to the databases of $\mathbb Z_4$ codes by Aydin {\em et al.}, at least nine new quaternary codes are found. Thanks to the special structure of the defining sets, we have the ability to determine whether the Gray images of certain obtained quaternary codes are linear or not. This allows us to obtain two infinite families of binary nonlinear codes and one infinite family of binary minimal linear codes. Furthermore, utilizing these minimal binary codes, some secret sharing schemes as a byproduct also are established.

翻译：摘要：近期，Hyun等人利用单纯复形构造了多族无限类极小最优线性码。基于其研究成果，本文通过单纯复形在环$\mathbb{Z}_4$上构造了更多线性码。具体而言，确定了所得四元码的Lee重量分布，并获得了两个无限类四重量四元码。与Aydin等人建立的$\mathbb{Z}_4$码数据库相比，发现至少九个新四元码。凭借定义集的特殊结构，我们得以判定某些所得四元码的Gray像是否为线性码。由此获得两个无限类二元非线性码及一个无限类二元极小线性码。此外，利用这些极小二元码，还附带建立了若干秘密共享方案。