New scattered linearized quadrinomials

Let 1 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.