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

• There are 13 classes of near-bent functions of degree less or equal to 3 modulo RM(1,7), the total number of such Boolean functions is 17158683162880 = approx=2^43.96. The fixator distribution is in this file.
• There are 4243469 classes of near-bent functions of degree greater than 3. The order of fixators take 50 distinct values and the total number of such Boolean functions is 88624918537535724072960 =approx= 76.23. .
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.