Whitney Numbers of Rank-Metric Lattices and Code Enumeration
Giuseppe Cotardo, Alberto Ravagnani, Ferdinando Zullo
TL;DR
The paper addresses the problem of computing the Whitney numbers of the first kind for rank-metric lattices $\mathscr{L}_i(n,m;q)$, linking these invariants to the open task of enumerating rank-metric codes with prescribed parameters. It develops a novel approach that fuses finite geometry (hyperovals and linear sets) with the linearized-polynomial description of rank-metric codes, producing new exact values and recursive tools for $w_j(i,n,m;q)$. Key contributions include a classification of dimension-2 $\mathbb{F}_{q^m}$-linear MRD codes, exact counts in the cases $q=2$ and $m=4$ with $q>2$, and density analyses of MRD codes in small dimensions. The work also provides a recursive framework for Whitney numbers, with explicit closed forms in low dimensions, highlighting the deep connection between MRD-code structures and scattered linear sets. Overall, the results advance both code enumeration and density questions in rank-metric settings, and demonstrate the power of finite-geometry methods in algebraic coding theory.
Abstract
We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.