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 $S(i,j)$ depend only on $i$ 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. 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.

翻译：布尔超立方体$Q_n$的完美$k$着色是一个从长度为$n$的二进制词集合到$k$种颜色集合上的函数，使得对于任意颜色$i$和$j$，每个颜色为$i$的词恰好有$S(i,j)$个颜色为$j$的邻居（汉明距离为$1$），其中系数$S(i,j)$仅取决于$i$和$j$，而与词的具体选择无关。所有系数$S(i,j)$构成的$k \times k$表格称为商矩阵。我们刻画了$Q_n$上度数不超过$3$的完美着色，即其商矩阵的所有特征值不小于$n-6$，或者等价地，每种颜色对应一个由$\mathbb{R}$上至多$3$次多项式表示的布尔函数。此外，我们刻画了$Q_n$上$(n-4)$阶相关免疫完美着色，其所有颜色对应$(n-4)$阶相关免疫布尔函数，或者等价地，商矩阵的所有非主特征值（不同于$n$的特征值）不大于$6-n$。关键词：完美着色，均衡划分，弹性函数，相关免疫函数。