New maximum scattered linear sets of the projective line

Abstract In [2] and [18] are presented the first two families of maximum scattered F q -linear sets of the projective line PG ( 1 , q n ) . More recently in [22] and in [5] , new examples of maximum scattered F q -subspaces of V ( 2 , q n ) have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the F q -linear sets presented in [22] and in [5] , for n = 6 , 8 , are new. Also, for q odd, q ≡ ± 1 , 0 ( mod 5 ) , we present new examples of maximum scattered F q -linear sets in PG ( 1 , q 6 ) , arising from trinomial polynomials, which define new F q -linear MRD-codes of F q 6 × 6 with dimension 12, minimum distance 5 and left idealiser isomorphic to F q 6 .