MRD-codes arising from the trinomial x^q+x^{q^3}+cx^{q^5}in F_{q^6}

In [10], the existence of Fq-linear MRD-codes of Fq 6×6, with dimension 12, minimum distance 5 and left idealiser isomorphic to Fq6, defined by a trinomial of Fq6[x], when q is odd and q≡0,±1(mod5), has been proved. In this paper we show that this family produces Fq-linear MRD-codes of Fq 6×6, 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 Fq-linear sets of PG(1,q6) are not PΓL(2,q6)-equivalent to any previously known linear set.