论文标题
等级晶格
Rank-Metric Lattices
论文作者
论文摘要
我们介绍了等级几何晶格的类别，并启动其结构特性的研究。排名式的晶格可以看作是$ Q $ - Q $ - 重量大道的晶格的动物，这是由1971年Dowling自己定义的。我们充分表征了可超过的排名式晶格，并计算了其特征性的多项式。然后，我们专注于最小的等级晶格，其特征多项式无法计算，并在其第一类的惠特尼数字上为其提供公式。证据依赖于计算结果和矢量排名码的理论，我们从等级 - 晶格的角度在本文中回顾了这一点。更确切地说，我们介绍了等级代码的晶格级权重的概念，并研究了它们作为组合不变的属性，并作为非等价代码的代码区分。
We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on the smallest rank-metric lattice whose characteristic polynomial we cannot compute, and provide a formula for it under a polynomiality assumption on its Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.
