Classification of Boolean Quartics Forms in eight Variables

This page reports the numerical experiments related to the classification of the Boolean quartic forms in eight variables that we computed (G. Leander, P. Rabizzoni, P. Veron, J.P. Zanotti and I) in 2006-07. Using Xiang-Dong Hou 's works, there are 999 Boolean quartic forms in eight variables up to GL(2,8) equivalence. We obtained the complete classification on the 9th july 2007. We used the multiplicative invariants, a quadratic invariant, and one Fourier Lift by derivation. These invariants split the space of quartic forms in 966 class, with 27 collisions of order 2, and 3 collision of order 3.

See the Slides of the Zvenigorod conference, September 2007, for details.

## Covering Radius of RM(3,8)

As expected, the classification of RM(4,8)* gives new results concerning the covering radius of the third order Reed-Muller RM(3,8) of length. Using some recent tricks of Claude Carlet, it is possible to show that the distance from the quartic
Q=2345+1246+1356+2467+3467+2567+1348+1258+1358+2478+3578+1678
to RM(3,8) is greater or equal to 50. This is an improvement for the tables of the Handbook of coding theory where the estimation of Xiang-Dong Hou claims this covering radius in the range 44-67.

Note that Ilya Dumer kindly accepted to run a decoding algorithm for us in order to decode this function. The computation shows that the distance is smaller or equal to 52.

## Counting bent functions

There are 536 class of quartic forms Q (header) providing bent functions of the form Q+f where f is a cubic functions. The computation of all bent functions finished on 31 December 2007, we obtained :
193887869660028067003488010240 = approx = 2^97.29
bent functions up to affine terms. This means the probability for a quartic function to be bent is around 2^{-57}. The reductions of the total number Bent functions modulo the small odd primes dividing the order of GL(2,8) i.e. 3, 5, 7, 17, 31 and 127 are :
Reduction modulo 3 : 2
Reduction modulo 5 : 0
Reduction modulo 7 : 0
Reduction modulo 17 : 13
Reduction modulo 31 : 0
Reduction modulo 127 : 0
as expected by the theory.
Philippe Langevin, Last modification on December 31th 2007.
Correction on 4 july 2008.