# The best linear approximation

Good afternoon. I'm interesting in finding the best (or maybe good) linear approximation of the function $F^{(19)}: V_{56}\to V_8$, where $$\begin{array}{l} F^{(1)}(x_1,x_2,\ldots, x_7) = P(x_2+x_6); \ F^{(2)}(x_1,\ldots,x_7) = F^{(1)}(x_2,\ldots,x_7,F^{(1)}(x_1,\ldots,x_7)), \ F^{(3)}(x_1,\ldots,x_7) = F^{(1)}\left(x_3,\ldots,x_7, F^{(1)}(x_1,\ldots,x_7), F^{(2)}(x_1,\ldots, x_7) \right) \ldots \end{array}$$ Here $x_j\in V_8$, $P$ is a byte permutation and by $+$ we denote addition in $V_8$.

Maybe there are methods for finding not best but good linear approximations for such a function? Thank you.