  In this paper we consider a family $\mathcal{F}$ of $16$-dimensional
$\mathbb{F}_q$-linear rank metric codes in $\mathbb{F}_q^{8\times8}$, arising
from the polynomial $x^{q^s}+\delta x^{q^{4+s}}\in\mathbb{F}_{q^8}[x]$.
Examples of MRD codes in $\mathcal{F}$ have been provided by Csajb\'ok, Marino,
Polverino and Zanella (2018). For any large enough odd $q$, we determine
exactly which codes in $\mathcal{F}$ are MRD. We also show that the MRD codes
in $\mathcal{F}$ are not equivalent to any other MRD codes known so far.
