  In this note, we go further on the "basis exchange" idea presented in
\cite{LiNa1} by using Mobious inversion. We show that the matrix
$S_1(f)S_0(f)^{-1}$ has a nice form when $f$ is chosen to be the majority
function, where $S_1(f)$ is the matrix with row vectors $\upsilon_k(\alpha)$
for all $\alpha \in 1_f$ and $S_0(f)=S_1(f\oplus1)$. And an exact counting for
Boolean functions with maximum algebraic immunity by exchanging one point in
on-set with one point in off-set of the majority function is given.
Furthermore, we present a necessary condition according to weight distribution
for Boolean functions to achieve algebraic immunity not less than a given
number.
