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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 leftidealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of$\mathbb{F}_{q^6}[x]$, when $q$ is odd and $q\equiv 0,\pm 1\pmod 5$, has beenproved. In this paper we show that this family produces $\mathbb{F}_q$-linearMRD-codes of $\mathbb{F}_q^{6\times 6}$, with the same properties, also in theremaining $q$ odd cases, but not in the $q$ even case. These MRD-codes are notequivalent to the previously known MRD-codes. We also prove that thecorresponding maximum scattered $\mathbb{F}_q$-linear sets of$\mathrm{PG}(1,q^6)$ are not $\mathrm{P}\Gamma\mathrm{L}(2,q^6)$-equivalent toany previously known scattered linear set.

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