"There are 536 class of quartic forms Q (header) [in 8 boolean variables] providing bent functions of the form Q+f where f is a cubic functions." Philippe Langevin, 2008.

What is the current prospect of enumerating the extended affine equivalence classes of the bent functions of degree 4 in 8 boolean variables? Is the problem hopelessly large? Has anyone made an attack on it? Are the methods used by Hall in her thesis still the state of the art? Could the methods used by Langevin et al. contribute to such an attack?