# Generate $4\times 4$ Sboxes with a given branch number

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?

One way to construct S-box with branch number 3 is to use concept of differential-equivalence, DDT-equivalence and the $$\gamma$$-equivalence as shown in paper "Two Notions of Differential Equivalence on Sboxes".
The new paper accepted "Reconstructing an S-box from its Difference Distribution Table " provides algorithm to reconstruct S-box . therefore, you could apply the concept on $$\gamma$$-equivalence DDT to find S-Box with branch number as you want.