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 4243482 classes of near-bent functions in seven variables modulo RM(1,7) and the total number of such Boolean functions 88624918554694407235840 =approx= 76.23. The detail of the classification is given in two files.
  • The file-one contains informations about the near bents of the form F+q where F in B(3,4,7) has trivial stabilizer, and three lines like : 
    anf=abc+acd+bcd+abe+ace+bce+bde+bcde+abcf+adf+bdf+cdef+beg+cdeg+fg+acfg+cdfg+aefg
stabSize=0
#nearbent=83
    means there are 83 classes of nearbents of the form F+q with q in RM(2,7)/RM(1,7). One have to run a program to recover such quadratic parts... That represents 3731797 class and a total of 78266228744129064468480 near bents modulo RM(1,7).
  • The file-two 
    anf=ac+bc+bd+abd+ae+abce+acde+cf+bcdf+ef+acef+bcef+adef+bdef+cg+bcg+eg+aeg+beg+fg
[1100111 0111011 0000100 0001000 0010000 0001010 0000001]0001110
[1100111 0111100 0000101 0001000 0010001 0001011 0000001]0000000
stabSize=8
    decribes an orbit having a stabilizer of order 8. There are 479545 class having trivial stabilizer. That represents 511685 class and a total of 10358689810565342767360 near bents modulo RM(1,7). The distribution of the orders of the stabilizers is summarized in this file.

    • Bent functions

    From this data one may probably count the number of bent functions in dimension 8.
    Philippe Langevin, Last modification February 2012, October 2022.