New scattered linearized quadrinomials

Let 18 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 phi(m,q)J=Xq(J(t-1)) +Xq(J(2t-1))+m(Xq(J)-Xq(J(t+1))) is an element of F-q(2t)[X], q odd, t >= 3 is R-q(t)-partially scattered for every value of m is an element of F-qt* and J coprime with 2t. Moreover, for every t>4 and q>5 there exist values of m for which phi(m,q) is scattered and new with respect to the polynomials mentioned in (i), (ii) and (iii) above. The related linear sets are of Gamma L-class at least two.