MRD-codes arising from the trinomial $x^q+x^{q^3}+c{x}^{q^5}\in {\mathbb{F}}_{q^6}[x]$

In [10], the existence of ${\mathbb{F}}_q$-linear MRD-codes of ${\mathbb{F}}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left idealiser isomorphic to ${\mathbb{F}}_{q^6}$, defined by a trinomial of ${\mathbb{F}}_{q^6}[x]$, when $q$ is odd and $q\equiv 0,\pm 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$, has been proved. In this paper we show that this family produces ${\mathbb{F}}_q$-linear MRD-codes of ${\mathbb{F}}_q^{6\times 6}$, with the same properties, also in the remaining $q$ odd cases, but not in the $q$ even case. These MRD-codes are not equivalent to the previously known MRD-codes. We also prove that the corresponding maximum scattered ${\mathbb{F}}_q$-linear sets of $\mathrm{P}\mathrm{G}(1,q^6)$ are not $\mathrm{P}\mathrm{\Gamma }\mathrm{L}(2,q^6)$-equivalent to any previously known scattered linear set.