MRD codes with maximum idealizers

TL;DR

This paper classifies MRD codes with maximum idealizers in small dimensions, revealing new families beyond generalized Gabidulin codes and establishing their limitations for larger sizes.

Contribution

It provides a complete classification of MRD codes with maximum idealizers for dimensions up to 9, identifying new code families and their connection to Moore-type matrices.

Findings

New MRD code families for n=7, q odd and n=8, q ≡ 1 mod 3.

Classification of MRD codes with maximum idealizers for n ≤ 9.

No additional such codes exist for n ≥ 9.

Abstract

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in $\mathbb{F}_q^{n\times n}$ the idealizers have been proved to be isomorphic to finite fields of size at most $q^n$. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in $\mathbb{F}_q^{n\times n}$ for $n\leq 9$ with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealizers for $n=7$, $q$ odd and for $n=8$, $q\equiv 1 \pmod 3$. These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for $n\geq 9$.