Differentially 4-uniform permutation

Functions	Conditions	References
$x^{2^{i}+1}$	$gcd(i,n)=2,n=2t$ and t is odd
$sx^{q+1}+x^{2^{i}+1}+x^{q(2^{i}+1)}+cx^{2^{i}q+1}+c^{q}x^{2^{i}+q}$	$q=2^{m},n=2m,$ $gcd(i,m)=1$, $c\in \mathbb {F} _{2^{n}},s\in \mathbb {F} _{2^{n}}\setminus \mathbb {F} _{q},X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1{\text{ has no solution }}x$ s.t. $x^{q+1}=1$
$x^{3}+a^{-1}\mathrm {Tr} _{n}(a^{3}x^{9})$	$a\neq 0$
$x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{3}x^{9}+a^{6}x^{18})$	$3|n$, $a\neq 0$
$x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{6}x^{18}+a^{12}x^{36})$	$3|n,a\neq 0$
$ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1}x^{2^{s}+2^{k+s}}$	$n=3k,\gcd(k,3)=\gcd(s,3k)=1,v,w\in \mathbb {F} _{2^{k}},vw\neq 1,3|(k+s),u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}$
$(x+x^{2{^{m}}})^{2^{k}+1}+u'(ux+u^{2^{m}}x^{2^{m}})^{(2^{k}+1)2^{i}}+u(x+x^{2^{m}})(ux+u^{2^{m}}x^{2^{m}})$	$n=2m,m\geqslant 2$ even, $\gcd(k,m)=1$ and $i\geqslant 2$ even, $u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*},u'\in \mathbb {F} _{2^{m}}{\text{ not a cube }}$
$a^{2}x^{2^{2m+1}+1}+b^{2}x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^{2}+c)x^{3}$	$n=3m,m\ {\text{odd}},L(x)=ax^{2^{2m}}+bx^{2^{m}}+cx$ satisfies the conditions in Lemma 8 of [7]