Explicit reduced APN permutations: ramification and optimal Galois-ring lifts

Let $m\ge5$ be odd and $q=2^m$. For every odd $r$ with $3\le r<m$ and $\gcd(r,m)=1$ we give the reduced polynomial $P_{m,r}=\mathrm{red}((T_r^3)^{2^{m-2}})$, $T_r=\sum_{i<r}X^{2^i}$, explicitly: it has coefficients in $\mathbb{F}_2$ and exactly $3r-2$ terms, it is an APN permutation of $\mathb