Covering radius of RM(4,8)
This page reports the numerical experiments related to the classification of the Boolean Functions in 8 variables that we computed in Autumn 2022. Mainly, we use a part of the classification obtained in 7 variables to classify the space RM(6,8)/RM(4,8) from which we deduce novelties about the covering radius of RM(4,8).
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 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| 1 | 9 | 3814830 | 1027.6 | 1044.5 | 1052.9 | 1055.3 | 1055.6 |
| 2 | 5 | 20748 | 1025.2 | 1042.0 | 1050.5 | 1052.9 | 1053.2 |
| 3 | 32 | 1016.7 | 1033.6 | 1042.0 | 1044.5 | 1044.8 | |
| 4 | 999 | 1016.7 | 1025.2 | 1027.6 | 1027.9 | ||
| 5 | 32 | 20748 | 3814830 | 7611801 | |||
| 6 | 5 | 9 | 14 | ||||
| 7 | 2 | 3 | |||||
| 8 | 2 |
The objective of this numerical experiment consists in providing the classifications of B(5,6,8) = RM(6,8)/RM(4,8) to deduce the covering radius of RM(4,8).