New scattered linearized quadrinomials

Let 1 < t < n be integers, where t is a divisor of n.An R-q tpartially scattered polynomial is a q-polynomial f in F q n [X] that satisfies the condition that for all x, y ∈ F * q n such that x/y ∈ F q t , if f (x)/x = f (y)/y, then x/y ∈ F q ; f is called scattered if this implication holds for all x, y ∈ F * q n .Two polynomials in F q n [X] are said to be equivalent if their graphs are in the same orbit under the action of the group ΓL(2, q n ).For n > 8 only three families of scattered polynomials in F q n [X] are known: (i) monomials of pseudoregulus type, (ii) binomials of Lunardon-Polverino type, and (iii) a family of quadrinomials defined in [1,10] and extended in [8,13].In this paper we prove that the polynomial ϕ m,q J = X q J(t-1) + X q J(2t-1) + m(X q J -X q J(t+1) ) ∈ F q 2t [X], q odd, t ≥ 3 is R-q tpartially scattered for every value of m ∈ F * q t and J coprime with 2t.Moreover, for every t > 4 and q > 5 there exist values of m for which ϕ m,q is scattered and new with respect to the polynomials mentioned in (i), (ii) and (iii) above.The related linear sets are of ΓL-class at least two.