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

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