Classification of RM(3,8)/RM(1,8)

This page reports a numerical experiment suggested by Augustine Muskuwa related to the classification of RM(3,8) modulo RM(1,8). Done on Automn 2018, the running time for this job was about 1 hour.

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 linear group GL(2, m) acts over the spaces RM(k,m), and thus on RM(k,m)/RM(s,m) when s<=k. Two Boolean functions f and g are said to be equivalent modulo RM(k, m) if there exists a linear transformation A in GL(2,m) such that

g = f o A modulo RM(k,m)

Affine and Linear group

The linear and affine groups have order :

#GL(2, m) = (2^m-1)(2^m-2)...(2^m-2^(m-1) ) #GA(2, m) = 2^m #GL(2,m)

These groups acts over RM(k,m) and also over RM(k,m)/RM(s,m). Let nl(m, s, t) and n(m,s,t) the ranks of the actions.

Objectives

The main objective of this numerical experiment is to provide the datas corresponding to the classification of RM(3,8)/RM(1,8), playing with Burnside's formula we get

nl(8,1,3) = 3796971 classes.

Methodology

Nothing special, we use the method described here starting from the full classification of the 32 classes of RM(3,8)* under the GL-action to build the file
[ gl-1-3-8.txt ]
that contains the 3796971 classes of RM(3,8)/RM(1,8) under the GL-action.

Affine action

In the same way, we get :
[ ag-1-3-8.txt ]
We found n(8,1,3) = 20748 members, the value has to be checked.

Application

It is well known that APN-permutation of degree 3 does not exist in dimension 6. Applying L4-norm filter one can see that a pure cubic APN-permutation in dimension must have 85 components in the 8 class : 
---
num=1
anf=ab+bc+acd+de+cf+ef+aef+bcg+dg+ch+agh
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 8 [-32], 88 [-16], 48 [0], 104 [16], 8 [32],
Correlation  : 2 [-64], 16 [-32], 219 [0], 16 [32], 2 [64], 1 [256],
---
num=2
anf=ac+bc+abc+bd+abd+ae+be+abe+bf+cf+df+abg+cg+beg+ach+adh+fgh
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 8 [-32], 88 [-16], 48 [0], 104 [16], 8 [32],
Correlation  : 24 [-32], 219 [0], 8 [32], 4 [64], 1 [256],
---
num=3
anf=ab+ac+abc+abd+be+abe+de+cf+ag+abg+beg+bh+ach+adh+fgh
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 8 [-32], 88 [-16], 48 [0], 104 [16], 8 [32],
Correlation  : 24 [-32], 207 [0], 24 [32], 1 [256],
---
num=4
anf=abd+be+ade+cde+cf+bcg+aeg+ah
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 4 [-32], 96 [-16], 48 [0], 96 [16], 12 [32],
Correlation  : 243 [0], 12 [64], 1 [256],
---
num=5
anf=abd+bcd+de+cf+bg+aeg+ah
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 8 [-32], 88 [-16], 48 [0], 104 [16], 8 [32],
Correlation  : 6 [-64], 243 [0], 6 [64], 1 [256],
---
num=6
anf=ad+abd+cd+ce+bf+beg+ah+cgh
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 4 [-32], 96 [-16], 48 [0], 96 [16], 12 [32],
Correlation  : 243 [0], 12 [64], 1 [256],
---
num=7
anf=abd+ae+ce+af+bf+dg+beg+ah+cgh
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 8 [-32], 88 [-16], 48 [0], 104 [16], 8 [32],
Correlation  : 2 [-64], 16 [-32], 219 [0], 16 [32], 2 [64], 1 [256],
---
num=8
anf=ad+abd+ae+be+af+bf+cf+dg+beg+ah+cgh
type=0 bal=1 deg=3 val=22 alpha=1.7500
Walsh spectrum : 8 [-32], 88 [-16], 48 [0], 104 [16], 8 [32],
Correlation  : 2 [-64], 16 [-32], 219 [0], 16 [32], 2 [64], 1 [256],
Waiting for applications !
Philippe Langevin, Last modification Autumn 2018.