The generalized Hamming weights (GHWs) of a linear code C extend the concept of minimum distance, which is the minimum cardinality of the support of all one-dimensional subspaces of C, to the minimum cardinality of the support of all r-dimensional subspaces of the code. In this work, we introduce Cartesian square-free codes, which are linear codes generated by evaluating square-free monomials over a Cartesian set. We use commutative algebraic tools, specifically the footprint bound, to provide explicit formulas for some of the GHWs of this family of codes, and we show how we can translate these results to evaluation codes over the projective space.

翻译：线性码C的广义汉明权重（GHWs）将最小距离的概念——即C的所有一维子空间支撑集的最小基数——推广至该码所有r维子空间支撑集的最小基数。本文引入笛卡尔无平方码，这是一类通过在笛卡尔集合上对无平方单项式求值生成的线性码。我们运用交换代数工具，特别是footprint界，为该码族的部分GHWs提供了显式公式，并展示了如何将这些结果推广至射影空间上的求值码。