Lower bounds on APN-distance for all known APN functions

The following tables list a lower bound on the Hamming distance between all known CCZ-inequivalent APN representatives up to dimension 11 using the methods described used in . Note that the lower bound is a CCZ-invariant (unlike the exact minimum distance itself) and it can be calculated via the formula $l(F)=\lceil {\frac {m_{F}}{3}}\rceil +1$, where $l(F)$ is the lower bound on the Hamming distance between an $(n,n)$-function $F$ and the closest APN function, and $m_{F}$ is defined as $m_{F}=\min _{b,\beta \in \mathbb {F} _{2^{n}}}|\{a\in \mathbb {F} _{2^{n}}:(\exists x\in \mathbb {F} _{2^{n}})(F(x)+F(a+x)+F(a+\beta )=b)\}|$. The values of $m_{F}$ for the CCZ-inequivalent representatives are provided in the tables for convenience. The representatives for dimensions 7 and 8 are taken from the list ofKnown quadratic APN polynomial functions over GF(2^7) and Known quadratic APN polynomial functions over GF(2^8), respectively, while the rest are taken from the table of CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11).

The tables for dimensions 7 and 8 can be found under Lower bounds on APN-distance for all known APN functions in dimension 7 and Lower bounds on APN-distance for all known APN functions in dimension 8, respectively, due to their large size.

DIMENSION 9
ID	$\Pi _{F}^{0}$	$m_{F}$	lower bound
1	255511, 512	255	86
2	255511, 512	255	86
3	255511, 512	255	86
4	255511, 512	255	86
5	255511, 512	255	86
6	255511, 512	255	86
7	2313, 23745, 24027, 24336, 24654, 24936, 25236, 25537, 25827, 26145, 26454, 26745, 27036, 2739, 27618, 2793, 512	231	78
8	255511, 512	255	86
9	255511, 512	255	86
10	255511, 512	255	86
11	255511, 512	255	86