Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations

TL;DR

This paper advances the understanding of Ferrers diagram codes by proving new cases where a key conjecture holds and analyzing the prevalence of maximal codes, especially MRD codes, as field size increases.

Contribution

It extends known cases of the Ferrers diagram conjecture and studies the asymptotic proportion of maximal Ferrers diagram codes, focusing on MRD codes.

Findings

Several new Ferrers diagrams confirmed to satisfy the conjecture.

The proportion of maximal codes approaches a limit depending on the Ferrers diagram.

For [m×2]-MRD codes with rank 2, the limiting proportion is approximately 1/e.

Abstract

This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, the proportion of maximal Ferrers diagram codes within the space of all rank-metric codes with the same shape and dimension is investigated. Special attention is being paid to MRD codes. It is shown that for growing field size the limiting proportion depends highly on the Ferrers diagram. For instance, for $[m\times 2]$-MRD codes with rank~$2$ this limiting proportion is close to $1/e$.