Say, I want to generate a lot of $4\times4$ Sboxes with linear (or differential) branch numbre of $3$. One idea is to take all the $302$ affine classes, expand each of those classes and check if any of the Sboxes has the desired branch number.
Is there any better way/known result (such as, this particular class does not contain any Sbox with linear branch number $3$) that can be used to reduce the search space?