Classification of Near Bent Functions in dimension 7


This page reports the numerical experiments related to the classification of near bent functions with 7 variables in February 2012, the total running time was about 4 days.


Definitions

Let m be a positive odd integer. A Boolean function f from GF(2)^m into GF(2) is said to be near bent if its Wash spectrum contains the three values : - 2^(m+1)/2, 0 or +2^(m+1)/2.

Objectives

The main objective is to classify the Near Bent functions in dimension 7 coming from bent functions in dimension 8 by starting from the classification of Boolean forms of degree 4 in 7 variables.

Methodology


A very naive approach works. For each quartic f in RM(4,7)/RM(2,7) satisfying the quadratic condition that must satisfy every the bent functions in dimension 8, we simply check which quadratic form q make f+q near bent.

Results

Finaly, we found 88624918554694407235840 near-bent functions in dimension 7 coming from bent functions, up to an affine term.

Bent functions

From this data one may probably count the number of bent functions in dimension 8.

Philippe Langevin, Last modification February 2012.