  Very recently, Tu et al. presented a sufficient condition about
$(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x) = x^{3\cdot 2^m} + a_1
x^{2^{m+1}+1}+ a_2 x^{2^m+2} + a_3 x^3$ is a class of permutation polynomials
over $\gf_{2^{n}}$ with $n=2m$ and $m$ odd. In this present paper, we prove
that the sufficient condition is also necessary.
