**Classification of Partial Spread Functions in eight Variables**

This page reports the numerical experiments related to the classification of
the Partial Spread Boolean Functions in eight variables that I computed
in 2008-2009. This work shows there are

70576747237594114392064 = approx = 2^{75.9}
bent functions in the PS classes of Dillon. The reader
will find some details in the
preprint
.
## Definitions and objectives

Let m = 2t be an even integer. A partial spread of order n is a
subset of n subspaces of dimension t in GF(2)^m. The linear group
GL(2,m) acts naturally over partial spreads. If f_1, f_2, ..., f_n
denote the indicator functions of the subspaces defining a partial
spread, the Boolean function f_1 + f_2 + ... + f_n (mod 2) is called
a partial spread function. When n = 2^t or n = 2^t+1, these functions
are bent. According to Dillon's terminology, they form respectively
the PS- and PS+ classes of bent functions. The main purpose of the
numerical experiment is to count the number of bent functions in
these classes for m = 8.
## Methodology

We proceed by induction, buiding a set of representatives of partial
spread functions of order (n+1) from a set of representative of partial
spread functions of order n.
- Extension.
- Pre-classification.
We apply invariant of forms of degree four to split the collection
of partial spread functions in 966 lists of partial spread functions.
- Classification.
Stabilization.

## Running times

extension classification stabilization
n time size time time class time psf
4 1 5 1 0 3 1 64374841666437120
5 15 233 55 10 22 10 20267057123180937216
6 69 4893 1162 385 341 6 1339989812392369324032
7 415 29691 7038 7246 3726 62 17833337132662061531136
8 1076 60943 14449 33501 9316 229 46056096661467073413120
9 681 31715 7516 8594 5442 19529 24520650576127040978944
10 219 8871 2109 698 1336 23 4731497045822911021056
11 75 2759 654 148 303 6 713809537614313684992
12 20 675 160 30 42 10 38019657690425327616
13 3 96 23 4 6 2 129740065512357888
14 0 11 3 0 1 59 44213490155520
15 0 3 1 0 1 11186 6579388416
16 0 2 0 0 1 0 200787
17 0 1 0 0 1 0 1

## file of representatives

For each n, the file psf-n.txt contains the representatives
of partial spread functions of order n. The datas are presented
as follow :
psf = (identifier) : (n-3) 16-bits numbers
fix = (order of the fixator group).
anf = (algebraic form)
rnk = private
spec= (spectrum)

All the partial spreads of order n that we computed are extensions
of the standard partial spread of order 3 : [ I: 0 ] [ 0 : I] [ I : I ]. The
information on line psf= describes the generator matrices [ I : A ] of the
other subspaces. A 16-bits number (A3 A2 A1 A0) represents a 4x4 matrix whose
the rows are A0, A1, A2, A3.
## Distribution of fixator sizes

## Partial spread of low degree

The degree of a partial spread function is less or equal to 4. It is
interesting to notice there are 2 ( and only 2 ! ) classes of n-spread
with degree less than 4 ( n < 15 ). They ly both in the PS+ class. One is a cubic :
psf-9.txt:rnk=999
psf-9.txt-fix=1008
psf-9.txt-anf=+25+135+45+345+126+36+136+46+246+346+27+127+137+47+147+247+347+18+28+38+138+348+358+268+368+278+378+478
psf-9.txt-spec= 136 [-16], 120 [16]

and the other one is quadratic :
psf-9.txt-psf=5442 : 8137 27484 30180 33825 39544 57749 65213
psf-9.txt:rnk=999
psf-9.txt-fix=348364800
psf-9.txt-anf=+25+16+36+46+27+47+28+38
psf-9.txt-spec= 136 [-16], 120 [16],

## Conclusion

We know that the number of bent functions is 2^{97.3}. This
computation shows that most of the bent functions do not ly
in one of the known classes : partial spread of Dillon or
Maiorana-Mcfarland.
Philippe Langevin, Last modification on April 4th 2010.