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>l(F) = \lceil \frac{m_F}{3} \rceil + 1</math>, where <math>l(F)</math> is the lower bound on the Hamming distance between an <math>(n,n)</math>-function <math>F</math> and the closest APN function, and <math>m_F</math> is defined as <math>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>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 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>\Pi_F^0</math>	<math>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

DIMENSION 10
ID	<math>\Pi_F^0</math>	<math>m_F</math>	lower bound
1	495682, 543341, 1024	495	166
2	495682, 543341, 1024	495	166
3	495682, 543341, 1024	495	166
4	47740, 48330, 48960, 49557, 501220, 50770, 513160, 519105, 525150, 53140, 53750, 5431, 54940, 1024	477	160
5	495682, 543341, 1024	495	166
6	495682, 543341, 1024	495	166
7	495682, 543341, 1024	495	166
8	495682, 543341, 1024	495	166

DIMENSION 11
ID	<math>\Pi_F^0</math>	<math>m_F</math>	lower bound
1	10232047, 2048	255	86
2	10232047, 2048	255	86
3	10232047, 2048	255	86
4	10232047, 2048	255	86
5	10232047, 2048	255	86
6	10232047, 2048	255	86
7	10232047, 2048	255	86
8	10232047, 2048	255	86
9	10232047, 2048	255	86
10	10232047, 2048	255	86
11	10232047, 2048	255	86
12	97811, 98133, 98422, 98766, 99066, 99355, 99677, 99977, 1002110, 100555, 100866, 101155, 1014121, 101755, 1020121, 102378, 102688, 102988, 103266, 1035110, 103855, 104199, 104477, 104788, 105044, 105366, 105655, 105944, 106266, 106522, 106811, 2048	978	326
13	10232047, 2048	255	86