Counting first-order correlation-immune functions
We first show that the exact number of $1$-resilient boolean functions with $7$ variables is $23478015754788854439497622689296$ and we obtain a tight estimation of their number with $8$ variables, between $4\;10^{67}$ and $5.6\;10^{68}$. We then present a general lower bound for the number of $1$-resilient boolean functions and improve Schneider's upper bound. We also propose a general lower bound for the number of $k$-resilient functions. Most of the bounds presented in this paper, substantially improve the best known bounds in the literature. We finally establish that the probability of a Boolean function being $1$-resilient is asymptotically between $\frac{\left(n\pi\right)^{n/2}}{2^{n^2-\frac{3}{2}n-1}e^{n-1/2}}$ and $2^{-\frac{n^2}{4}+\frac{n}{4}}$. We also present some ideas about how we may be able to find the exact asymptotic value of this probability.
This is joint work with Jean-Marie Le Bars from University of Caen.