Whitney Numbers of Rank-Metric Lattices and Code Enumeration

TL;DR

This paper studies the Whitney numbers of rank-metric lattices, connecting them to the enumeration of rank-metric codes, and provides asymptotic estimates for code density functions using hyperoval and linear set methods.

Contribution

It introduces new methods to compute Whitney numbers for infinite families of rank-metric lattices and derives asymptotic estimates for code densities.

Findings

Computed Whitney numbers for specific rank-metric lattice families

Provided asymptotic estimates for the density of certain rank-metric codes

Linked lattice combinatorics to code enumeration problems

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.