On symplectic matrices of cubic Boolean forms and connections with second order nonlinearity

The best quadratic approximations of cubic Boolean functions are studied in this paper. By exploiting recent results on the classification of Boolean functions, we introduce the notion of symplectic matrices of cubic forms and show the special structure obtained by forms of almost maximum distance from all quadratic functions. These results lead to new lower bound on the covering radius of R(2, n) in R(3, n), i.e. the second order nonlinearity of cubic functions, better than the Cohen et al. bound for n les 15.