Rank-Metric Codes, Semifields, and the Average Critical Problem

TL;DR

This paper explores the density of MRD codes in the rank metric and connects it to the Critical Problem in combinatorial geometry, providing bounds and asymptotic analyses for various code families.

Contribution

It introduces new lower bounds for the density of MRD codes using semifield theory and links the problem to the classical Critical Problem, offering asymptotic insights.

Findings

Lower bounds for MRD code density are sharp for prime matrix sizes with large fields.

Binary field bounds are established for MRD code density.

Optimal codes in symmetric, alternating, and Hermitian contexts are shown to be sparse.

Abstract

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.