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.

翻译：本文研究秩度量格的一类Whitney数，该问题与具有特定参数的秩度量码的枚举这一公开问题密切相关。我们运用超卵形线理论与线性集理论中的方法，计算了无穷族秩度量格的Whitney数。作为结果的应用，我们证明了先前研究中猜想的一类特定秩度量码密度函数的渐近估计。