# Constructions of optimal rank-metric codes from automorphisms of rational function fields

## TL;DR

This paper introduces a new class of automorphisms of rational function fields to construct optimal rank-metric codes, including generalized Gabidulin codes and Ferrers diagram codes, advancing the theory and providing new MRD code examples.

## Contribution

It presents novel automorphisms for rational function fields to construct optimal rank-metric codes, including codes not equivalent to known twisted variants and resolving a conjecture on Ferrers diagram codes.

## Key findings

- Constructed generalized Gabidulin codes over rational function fields.
- Derived MRD codes from these constructions that are not equivalent to existing codes.
- Settled a conjecture on Ferrers diagram rank-metric codes.

## Abstract

We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.