**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).

## Definitions

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

g = f o A modulo RM(r,m)
## Affine group

The affine group has order :

#AG(2, m) = 2^m (2^m - 1)(2^m - 2)...(2^m - 2^(m-1) )
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 :
n(s, t, m ) = n( m-t, m - s, m)
## Objectives

s/t | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

1 | 9 | 3814830 | 10^{27.6} | 10^{44.5} | 10^{52.9} | 10^{55.3} | 10^{55.6} |

2 | 5 | 20748 | 10^{25.2} | 10^{42.0} | 10^{50.5} | 10^{52.9} | 10^{53.2} |

3 | | 32 | 10^{16.7} | 10^{33.6} | 10^{42.0} | 10^{44.5} | 10^{44.8} |

4 | | | 999 | 10^{16.7} | 10^{25.2} | 10^{27.6} | 10^{27.9} |

5 | | | | 32 | 20748 | 3814830 | 7611801 |

6 | | | | | 5 | 9 | 14 |

7 | | | | | | 2 | 3 |

8 | | | | | | | 2 |

some class numbers of B(s,t,8)

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).

## Methodology - Result - Application

The details are given in the following
[ ALCOCRYPT slides ]
[ abstract ]
[ FQ15 slides ]
[ FQ15 abstract ]
[ article ]
[ data ]
[ clip ]

Valérie Gillot,
Philippe Langevin,

Institut Mathématiques de Toulon,

last modification : june 2023.