Quaternary codes and their binary images
Yansheng Wu, Chao Li, Lin Zhang, Fu Xiao
TL;DR
This work constructs quaternary linear codes over $\mathbb{Z}_4$ by defining sets $D=\Delta_A+2\Delta^*$ derived from simplicial complexes with two maximal elements $A,B$, and analyzes their Lee weight distributions to produce two infinite families of four-Lee-weight codes. It further studies the Gray images under the Gray map, establishing when these images are binary linear or nonlinear, and identifies two binary nonlinear families and one binary minimal linear family, enabling secret sharing schemes. The paper proves exact Lee weight distributions, provides explicit parameter tables, and demonstrates that at least nine new $\mathbb{Z}_4$ codes arise beyond existing databases. These results expand the catalog of quaternary codes with well-characterized weight structures and reveal practical links to binary minimal codes and secret-sharing constructions, while suggesting broader poset-based generalizations for future work.
Abstract
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.