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 of Known 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 table below shows the lower bounds computed for some of the known APN functions for dimensions between 4 and 11. The functions in dimensions between 5 and 8 are indexed according to the table of Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8. The ones between 9 and 11 are index according to the table of CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11). A separate table listing these results for the (more than 8000) known APN functions in dimension 8 is available as Lower bounds on APN-distance for all known APN functions in dimension 8.

Dimension	F	[math]\displaystyle{ m_F }[/math]	Lower bound
4	x3	3	2
5	x3	15	6
5	x5	15	6
5	x15	9	4
6	1.1	27	10
6	1.2	27	10
6	2.1	15	6
6	2.2	27	10
6	2.3	27	10
6	2.4	15	6
6	2.5	15	6
6	2.6	15	6
6	2.7	15	6
6	2.8	15	6
6	2.9	21	8
6	2.10	21	8
6	2.11	15	6
6	2.12	15	6
7	7.1	54	19
7	all others	63	22
8	1.1 - 1.13	111	38
8	1.14	99	34
8	1.15 - 1.17	111	38
8	2.1	111	38
8	3.1	111	38
8	4.1	99	34
8	5.1	105	36
8	6.1	105	36
8	7.1	111	38
9	9.7	231	78
9	all others	255	86
10	10.4	477	160
10	all others	495	166
11	11.12	978	327
11	all others	1023	342