Binary additive MRD codes with minimum distance n-1 must contain a semifield spread set

TL;DR

This paper proves that certain additive MRD codes over GF(2) with minimum distance n-1 must include a semifield spread set, and classifies these codes for small n and over GF(3) using computational methods.

Contribution

It establishes a structural result linking additive MRD codes to semifield spread sets and provides a classification for small parameters both theoretically and computationally.

Findings

Additive MRD codes with minimum distance n-1 contain semifield spread sets in some cases.

Complete classifications of such codes for n ≤ 6 over GF(2) and in M_4(GF(3)).

MRD codes with minimum distance n-1 are rarer than those with distance n.

Abstract

In this paper we prove a result on the structure of the elements of an additive {\it maximum rank distance (MRD) code} over the field of order two, namely that in some cases such codes must contain a semifield spread set. We use this result to classify additive MRD codes in $M_n(\mathbb{F}_2)$ with minimum distance $n-1$ for $n\leq 6$. Furthermore we present a computational classification of additive MRD codes in $M_4(\mathbb{F}_3)$. The computational evidence indicates that MRD codes of minimum distance $n-1$ are much more rare than MRD codes of minimum distance $n$, i.e. semifield spread sets. In all considered cases, each equivalence class has a known algebraic construction.