# Automorphism groups and new constructions of maximum additive rank metric codes with restrictions

## TL;DR

This paper investigates the automorphism groups and equivalence classes of maximum additive rank metric codes, specifically symmetric, alternating, and hermitian matrix codes, and introduces a new maximum symmetric 2-code.

## Contribution

It determines automorphism groups and solves the equivalence problem for maximum additive rank metric codes with restrictions, and presents a novel maximum symmetric 2-code.

## Key findings

- Automorphism groups of symmetric, alternating, and hermitian maximum d-codes are characterized.
- The equivalence problem for these codes is solved.
- A new maximum symmetric 2-code not equivalent to known codes is constructed.

## Abstract

Let $d, n \in \mathbb{Z}^+$ such that $1\leq d \leq n$. A $d$-code $\mathcal{C} \subset \mathbb{F}_q^{n \times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $\mathcal{C}$, the rank of their difference is greater than or equal to $d$. A $d$-code with as many as possible elements is called a maximum $d$-code. The integer $d$ is also called the minimum distance of the code. When $d<n$, a classical example of such an object is the so-called generalized Gabidulin code. There exist several classes of maximum $d$-codes made up respectively of symmetric, alternating and hermitian matrices. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric $2$-code which is not equivalent to the one with same parameters known so far.