# On the Sparseness of Certain MRD Codes

## TL;DR

This paper investigates the rarity of certain MRD codes over finite fields, showing that as the field size grows, these codes become increasingly sparse within the space of all rank-metric codes.

## Contribution

It provides the first known example where MRD codes are proven to be sparse, using a novel reduction to bases and classical semifield results.

## Key findings

- MRD codes are sparse for parameters [3×3;3] as q increases
- Proportion of MRD codes tends to 0 when q→∞
- Reduction to basis spaces and semifield theory underpins the analysis

## Abstract

We determine the proportion of $[3\times 3;3]$-MRD codes over ${\mathbb F}_q$ within the space of all $3$-dimensional $3\times3$-rank-metric codes over the same field. This shows that for these parameters MRD codes are sparse in the sense that the proportion tends to $0$ as $q\rightarrow\infty$. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.