We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.
翻译:我们从半场理论中找出两个基本问题, 包括编码理论和组合几何, 重点是它们之间的关联。 这些问题是: 在标准等级中计算 MRD 代码的无症状密度问题, 以及 Crapo 和 Rota 组合式几何的关键问题。 我们从半场理论中从全位、 平方 MRD 代码的密度函数中得出两个较低的界限。 当矩阵大小是一个质数, 底部字段足够大时, 第一个界限是尖锐的, 而第二个界限则适用于二进制域。 我们然后从质量上, 通常是零位角度, 来重新审视组合式几何颜色的临界问题。 我们展示了这个非常古典化的问题与计算 MRD 代码的亚性密度之间的联系。 最后, 我们研究一些标准级代码中特殊组的无症状密度, 包括对称、 交替 和 Hermitian 。 我们特别显示这三种情况下的最佳代码是稀疏的 。