On perfect 2-colorings of the $q$-ary hypercube

A coloring of the $q$-ary $n$-dimensional cube (hypercube) is called perfect if, for every $n$-tuple $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean-valued function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-dimensional face of the hypercube. Let $f=\chi^S$ be a characteristic function of some subset $S$ of hypercube. In the paper the inequality $\rho(S)q(\operatorname{cor}(f)+1)\le A(S)$ is proved, where $\operatorname{cor}(f)$ is the maximum degree of the correlation immunity of $f$, $A(S)$ is the average number of neighbors in the set $S$ for $n$-tuples in a complement of a set $S$, and $\rho(S)=|S|/q^n$ is the density of the set $S$. Moreover, the function $f$ is a perfect coloring if and only if we obtain an equality in the above formula.