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 [math]\displaystyle{ l(F) = \lceil \frac{m_F}{3} \rceil + 1 }[/math], where [math]\displaystyle{ l(F) }[/math] is the lower bound on the Hamming distance between an [math]\displaystyle{ (n,n) }[/math]-function [math]\displaystyle{ F }[/math] and the closest APN function, and [math]\displaystyle{ m_F }[/math] is defined as [math]\displaystyle{ 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 ) \} | }[/math]. The values of [math]\displaystyle{ m_F }[/math] 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	[math]\displaystyle{ \Pi_F^0 }[/math]	[math]\displaystyle{ m_F }[/math]	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