On a family of linear MRD codes with parameters $[8\times8,16,7]_q$

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 $x^{qs}+\delta x^{{q{4+s}}\in\mathbb{F}_{q8}[x]$.} 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.