New scattered linearized quadrinomials

Let $1<t<n$ be integers, where $t$ is a divisor of $n$. An R-$q^t$-partially scattered polynomial is a ${\mathbb{F}}_q$-linearized polynomial $f$ in ${\mathbb{F}}_{q^n}[X]$ that satisfies the condition that for all $x,y\in {\mathbb{F}}_{q^n}^{\ast }$ such that $x\mathrm{/}y\in {\mathbb{F}}_{q^t}$, if $f(x)\mathrm{/}x=f(y)\mathrm{/}y$, then $x\mathrm{/}y\in {\mathbb{F}}_q$; $f$ is called scattered if this implication holds for all $x,y\in {\mathbb{F}}_{q^n}^{\ast }$. Two polynomials in ${\mathbb{F}}_{q^n}[X]$ are said to be equivalent if their graphs are in the same orbit under the action of the group $\mathrm{\Gamma }L(2,q^n)$. For $n>8$ only three families of scattered polynomials in ${\mathbb{F}}_{q^n}[X]$ 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 ${\phi }_{m,q^J}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^J}-X^{q^{J(t+1)}})\in {\mathbb{F}}_{q^{2t}}[X]$, $q$ odd, $t\ge 3$ is R-$q^t$-partially scattered for every value of $m\in {\mathbb{F}}_{q^t}^{\ast }$ 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 $\mathrm{\Gamma }L$-class at least two.