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A perfect $k$-coloring of the Boolean hypercube $Q_n$ is a function from theset of binary words of length $n$ onto a $k$-set of colors such that for anycolors $i$ and $j$ every word of color $i$ has exactly $S(i,j)$ neighbors (atHamming distance $1$) of color $j$, where the coefficient $S(i,j)$ depend onlyon $i$ and $j$ but not on the particular choice of the words. The $k$-by-$k$table of all coefficients $S(i,j)$ is called the quotient matrix. Wecharacterize perfect colorings of $Q_n$ of degree at most $3$, that is, withquotient matrix whose all eigenvalues are not less than $n-6$, or,equivalently, such that every color corresponds to a Boolean functionrepresented by a polynomial of degree at most $3$ over $R$. Additionally, wecharacterize $(n-4)$-correlation-immune perfect colorings of $Q_n$, whose allcolors correspond to $(n-4)$-correlation-immune Boolean functions, or,equivalently, all non-main (different from $n$) eigenvalues of the quotientmatrix are not greater than $6-n$.  Keywords: perfect coloring, equitable partition, resilient function,correlation-immune function.

On degree-$3$ and $(n-4)$-correlation-immune perfect colorings of $n$-cubes