# $\mathbb{F}_{q^n}$-linear rank distance codes and their distinguishers

## TL;DR

This paper surveys known $qn$-linear MRD-codes, studies their invariants, and characterizes generalized twisted Gabidulin codes, with implications for cryptography and decoding algorithms.

## Contribution

It provides a comprehensive survey of $qn$-linear MRD-codes, analyzes their invariants, and characterizes generalized twisted Gabidulin codes, advancing understanding of their structure and applications.

## Key findings

- Most rank distance codes are MRD for large fields
- Few $qn$-linear MRD-code families are known up to equivalence
- Characterization of generalized twisted Gabidulin codes

## Abstract

For any admissible value of the parameters there exist Maximum Rank distance (shortly MRD) $\mathbb{F}_{q^n}$-linear codes of $\mathbb{F}_q^{n\times n}$. It has been shown in \cite{H-TNRR} (see also \cite{ByrneRavagnani}) that, if field extensions large enough are considered, then \emph{almost all} (rectangular) rank distance codes are MRD. On the other hand, very few families of $\mathbb{F}_{q^n}$-linear codes are currently known up to equivalence. One of the possible applications of MRD-codes is for McEliece--like public key cryptosystems, as proposed by Gabidulin, Paramonov and Tretjakov in \cite{GPT}. In this framework it is very important to obtain new families of MRD-codes endowed with fast decoding algorithms. Several decoding algorithms exist for Gabidulin codes as shown in \cite{Gabidulin}, see also \cite{Loi06,PWZ,WT}. In this work, we will survey the known families of $\mathbb{F}_{q^n}$-linear MRD-codes, study some invariants of MRD-codes and evaluate their value for the known families, providing a characterization of generalized twisted Gabidulin codes as done in \cite{GiuZ}.