A new family of MRD-codes

We introduce a family of linear sets of $\mathrm{P}\mathrm{G}(1,q^{2n})$ arising from maximum scattered linear sets of pseudoregulus type of $\mathrm{P}\mathrm{G}(3,q^n)$. For $n=3,4$ and for certain values of the parameters we show that these linear sets of $\mathrm{P}\mathrm{G}(1,q^{2n})$ are maximum scattered and they yield new MRD-codes with parameters $(6,6,q;5)$ for $q>2$ and with parameters $(8,8,q;7)$ for $q$ odd.