Classification of Boolean functions in dimension 7
This page reports the numerical experiments related to the classification of the Boolean Functions in less than 7 variables that we computed in April 2022. In this continuation of the experiment, we provide more results and all the details presented at the BFA-2022 conference on Boolean functions and their applications.
Let m be a positive integer. The space of Boolean functions from GF(2)^m into GF(2) is denoted by RM(k,m). This notation comes from coding theory, it is the Reed-Muller code of order k in m variables. The affine group AG(2, m) acts over the spaces RM(k,m), and thus on RM(k,m)/RM(r,m) when r<=k. Two Boolean functions f and g are said to be equivalent modulo RM(r, m) if there exists an affine transformation A in AG(2,m) such that
The affine group has order :
it can be generated by three transformations : the shift S, a transvection T and a nontrivial translation U. The affine group acts over RM(k,m) and also over B(s,t,m) := RM(t,m)/RM(s-1,m). A formula for the rank n(s, t, m) of the action of AG(2,m) over B(s,t,m) was determined by Xiang-Dong Hou,
AGL(m, 2) Acting on R(r, m)/R(s, m), journal of algebra 171/3 (1995) It satisfies :| s/t | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| 0 | 3 | 12 | 3486 | 1013.5 | 1019.8 | 1021.9 | 1022.2 |
| 1 | 2 | 8 | 1890 | 1013.1 | 1019.5 | 1021.6 | 1021.9 |
| 2 | 4 | 179 | 10 11.0 | 1017.3 | 1019.5 | 1019.8 | |
| 3 | 12 | 68443 | 1011.0 | 1013.1 | 1013.5 | ||
| 4 | 12 | 179 | 1890 | 3486 | |||
| 5 | 4 | 8 | 12 | ||||
| 6 | 2 | 3 | |||||
| 7 | 2 |
The objective of this numerical experiment consists in providing the classifications of B(s,t,7) = RM(t,7)/RM(s-1,7) for all the reasonable parameters s and t.