How to generate a 16 X 8 perfectly nonlinear S-box
$A:\mathbb{F}_2^m\rightarrow \mathbb{F}_2^m$ is the state space map induced by the primitive length $m$ LFSR. 

Thus it maps the zero vector of length $m$ to itself and in general the LFSR state $(a_0,a_1, \ldots,a_{m-1})$ to $(a_1,\ldots,a_m)$.
$A^i$ is just $A$ iterated $i$ times (shift LFSR $i$ times).

Then, the APN map $z=(z_1,\ldots,z_n)=F(x_1,x_2)$ where $x_1,x_2,z,\in \mathbb{F}_2^m$ is given by
$$
z_i=A^{i-1}(x_1)\cdot x_2,\quad i=1,\ldots,m
$$
where $z_i$ are individual bits.