On the connection between correlation-immune functions and perfect 2-colorings of the Boolean n-cube

TL;DR

This paper explores the relationship between correlation-immune Boolean functions and perfect 2-colorings of the Boolean n-cube, establishing a key inequality that characterizes when such functions produce perfect colorings.

Contribution

It proves a new inequality linking correlation immunity, neighbor counts, and density, and characterizes perfect colorings via equality in this inequality.

Findings

Derived an inequality connecting correlation immunity and neighbor structure.

Characterized perfect colorings as cases where the inequality becomes equality.

Provided conditions under which Boolean functions induce perfect colorings.

Abstract

A coloring of the Boolean $n$-cube is called perfect if, for every vertex $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-face of the Boolean $n$-cube. In the present paper it is proven that each Boolean function $\chi^S$ ($S\subset E^n$) satisfies the inequality $${\rm nei}(S)+ 2({\rm cor}(S)+1)(1-\rho(S))\leq n,$$ where ${\rm cor}(S)$ is the maximum degree of the correlation immunity of $\chi^S$, ${\rm nei} (S)= \frac{1}{|S|}\sum\limits_{x\in S}|B(x)\cap S|-1$ is the average number of neighbors in the set $S$ for vertices in $S$, and $\rho(S)=|S|/2^n$ is the density of the set $S$. Moreover, the function $\chi^S$ is a perfect coloring if and only if we obtain an equality in the above formula. Keywords: hypercube, perfect coloring, perfect code, correlation-immune function.