  Left and right idealizers are important invariants of linear rank-distance
codes. In the case of maximum rank-distance (MRD for short) codes in
$\mathbb{F}_q^{n\times n}$ the idealizers have been proved to be isomorphic to
finite fields of size at most $q^n$. Up to now, the only known MRD codes with
maximum left and right idealizers are generalized Gabidulin codes, which were
first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and
Gabidulin in 2005. In this paper we classify MRD codes in
$\mathbb{F}_q^{n\times n}$ for $n\leq 9$ with maximum left and right idealizers
and connect them to Moore-type matrices. Apart from generalized Gabidulin
codes, it turns out that there is a further family of rank-distance codes
providing MRD ones with maximum idealizers for $n=7$, $q$ odd and for $n=8$,
$q\equiv 1 \pmod 3$. These codes are not equivalent to any previously known MRD
code. Moreover, we show that this family of rank-distance codes does not
provide any further examples for $n\geq 9$.
