The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However, determining the generalized Hamming weights of linear codes, especially the weight hierarchy, is generally challenging. In this paper, we investigate the generalized Hamming weights of a class of linear code $\C$ over $\bF_q$, which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in $\bF_q^m$. We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all $r$-dimensional subspaces of $\bF_q^m$, where $1\leq r\leq {\rm dim}_{\bF_q}(\C)$.

翻译：线性码的广义汉明重量研究是编码理论中的一个重要课题，因为它承载了码的结构信息并决定了其在不同应用中的性能。然而，确定线性码的广义汉明重量（特别是重量层级）通常极具挑战性。本文研究了一类基于定义集构造的$\bF_q$上线性码$\C$的广义汉明重量。这些定义集要么是特殊的单纯复形，要么是其在$\bF_q^m$中的补集。通过分析某些单纯复形与$\bF_q^m$所有$r$维子空间（其中$1\leq r\leq {\rm dim}_{\bF_q}(\C)$）之间的最大或最小交集，我们确定了这些码的完整重量层级。