In this paper, we construct three classes of permutation quadrinomials with Niho exponents of the form
$$f(x)=\alpha _0x^r+\alpha _1x^{s_1(p^m-1)+r}+\alpha _2x^{s_2(p^m-1)+r}+\alpha _3x^{s_3(p^m-1)+r}\in \mathbb {F}_{p^{n}}[x]$$
, where p is an odd prime,
$$n=2m $$
is a positive even integer, and
$$(r,s_1,s_2,s_3)=(1,\frac{-1}{p^k-2},1,\frac{p^k-1}{p^k-2})$$
,
$$(1,\frac{p^k+1}{p^k+2},1,\frac{1}{p^k+2})$$
and (3, 1, 2, 3), respectively. The exponents of the first two classes are considered for the fir...
