  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.

Combinatorics