On APN-functions and division property of multisets

In 2015, the division property was proposed as a tool to construct an integral distinguisher. According to this technique, the less the number $\left\lceil n/d \right\rceil $ is for a $n$-bit S-box of degree $d$, the fewer rounds might be in an integral distinguisher. In this paper, the number $\left\lceil n/d \right\rceil $ for some binary APN-transformations is studied. The best parameters of the APN-transformations are identified to reduce the number of rounds in the integral distinguisher.