& \chi ^S{}_{0,0,0,1,0} & \chi ^X{}_{1,1,0} & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,0,0,0} \\
L_3L_8 & \chi ^S{}_{0,0,0,1,0} & \chi ^S{}_{0,1,1,1,0} & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,0,0,0} \\
 L_4L_7& \chi ^S{}_{0,0,1,1,1} & \chi ^S{}_{0,1,0,1,1} & \chi ^X{}_{0,0,1} & \chi ^S{}_{0,1,1,0,0} \\
L_5L_6 & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,1,0,1} & \chi ^S{}_{0,0,1,1,0} & \chi ^X{}_{1,1,0} \\
L_5L_7 & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,1,0,0} & \chi ^S{}_{0,1,0,1,0} & \chi ^S{}_{0,0,1,1,0} \\
L_6L_8 & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,0,0,0} & \chi ^S{}_{0,0,0,1,0} & \chi ^S{}_{0,1,1,1,0} \\
L_7L_8 & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,1,1,0} & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,1,1,0} \\ 
\hline
\end{array}$

To illustrate (see Fig. \[flow\_map\]) that we choose the two-dimension face given by the local points $L_7$ and $L_8$ and employ the wiring presented in table \[2dvoid\_strategies\] (which is the same as the example shown for the one-dimensional scenario). In this case we have $B_0^0=B_0^7=B_7^0=B_0^8=PR$, $B_7^7=B_8^8=L_7$, $B_8^7=B_7^8=L_8$, and $B_8^0=\frac{1}{2}(L_7+L_8)$.

Given a box $p=c_{0}^{(0)} PR + c_7^{(0)}L_7+c_8^{(0)}L_8$, let $c_0^{(n)}$, $c_7^{(n)}$, and $c_8^{(n)}$ be the coefficients of $PR$, $L_7$, and $L_8$ respectively, after applying $n$ wirings. We have that the output $Q_n$ of the $n$-th wiring is: $$\begin{aligned}
\nonumber
& & q_n=\left[2-c_0^{(n-1)}-c_8^{(n-1)}\right]c_0^{(n-1)}PR+ \\  \nonumber
& & \frac{1}{2}\left[c^{(n-1)}_0c^{(n-1)}_8+ 2\left(c_7^{(n-1)}\right)^{2}+2\left(c_8^{(n-1)}\right)^{2}\right]L_7+ \\ \nonumber
& & \frac{1}{2}\left[c_0^{(n-1)}+4c_7^{(n-1)}\right]c_8^{(n-1)}L_8.\end{aligned}$$ Notice 