  Let $\mathcal{C}$ be a set of $m$ by $n$ matrices over $\mathbb{F}_q$ such
that the rank of $A-B$ is at least $d$ for all distinct $A,B\in \mathcal{C}$.
Suppose that $m\leqslant n$. If $\#\mathcal{C}= q^{n(m-d+1)}$, then
$\mathcal{C}$ is a maximum rank distance (MRD for short) code. Until 2016,
there were only two known constructions of MRD codes for arbitrary $1<d<m-1$.
One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was
later generalized by Kshevetskiy and Gabidulin (2005). We often call them
(generalized) Gabidulin codes. Another family was recently obtained by Sheekey
(2016), and its elements are called twisted Gabidulin codes. In the same paper,
Sheekey also proposed a generalization of the twisted Gabidulin codes. However
the equivalence problem for it is not considered, whence it is not clear
whether there exist new MRD codes in this generalization. We call the members
of this putative larger family generalized twisted Gabidulin codes. In this
paper, we first compute the Delsarte duals and adjoint codes of them, then we
completely determine the equivalence between different generalized twisted
Gabidulin codes. In particular, it can be proven that, up to equivalence,
generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets
of this family.
