Upper bounds on nonlinearity of correlation immune Boolean functions

It is known that $\mathrm{nl}(f)\le 2^{n-1}-2^m$ for the nonlinearity $\mathrm{nl}(f)$ of any Boolean function $f$ with $n$ variables and with the correlation immunity order $m$. We prove that for all $n\ge512$ and $0<m<n-1$, except two cases: $m=2^s$, $n=2^{s+1}+1$ and $m=2^s+1$, $n=2^{s+1}+2$ for $s\ge0$, this bound can be improved up to $\mathrm{nl}(f)\le2^{n-1}-2^{m+1}$. Besides, we have checked this result for $n<512$, $0<m<n-1$ using computer.