AbstractA class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the q-polynomial over $${{\mathbb {F}}}_{q^6}$$
F
q
6
, $$q \equiv 1\pmod 4$$
q
≡
1
(
mod
4
)
described in Bartoli et al. (ARS Math Contemp 19:125–145, 2020) and Zanella and Zullo (Discrete Math 343:111800, 2020) is generalized for any even $$n\ge 6$$
n
≥
6
to an $${{{\mathbb {F}}}_q}$$
F
q
-linear automorphism $$\psi (x)$$
ψ
(
x
)
of $${{\mathbb {F}}}_{q^n}$$
F
q
n
of order n. Such $$\psi (x)$$
ψ
(
x
)
and some functional powers of it are proved to be scattered. In particular, this provides new maximum scattered linear sets of the projective line $${{\,\mathrm{{PG}}\,}}(1,q^n)$$
PG
(
1
,
q
n
)
for $$n=8,10$$
n
=
8
,
10
. The polynomials described in this paper lead to a new infinite family of MRD-codes in $${{\mathbb {F}}}_q^{n\times n}$$
F
q
n
×
n
with minimum distance $$n-1$$
n
-
1
for any odd q if $$n\equiv 0\pmod 4$$
n
≡
0
(
mod
4
)
and any $$q\equiv 1\pmod 4$$
q
≡
1
(
mod
4
)
if $$n\equiv 2\pmod 4$$
n
≡
2
(
mod
4
)
.
Partially scattered linearized polynomials and rank metric codes