A Lower Bound on the Number of Boolean Functions with Median Correlation Immunity

TL;DR

This paper establishes a lower bound on the number of balanced correlation immune Boolean functions of order n/2, revealing their abundance grows super-exponentially with n.

Contribution

It provides the first asymptotic lower bound on the count of such Boolean functions, advancing understanding of their combinatorial complexity.

Findings

Number of balanced correlation immune functions grows super-exponentially with n

Asymptotic lower bound is n^{2^{(n/2)-2}(1+o(1))}

Results relate to orthogonal arrays and resilience in Boolean functions

Abstract

The number of $n$-ary balanced correlation immune (resilient) Boolean functions of order $\frac{n}{2}$ is not less than $n^{2^{(n/2)-2}(1+o(1))}$ as $n\rightarrow\infty$. Keywords: resilient function, correlation immune function, orthogonal array