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.

翻译：布尔超立方体$Q_n$的一个完美$k$着色是指从长度为$n$的二进制字集合到一个$k$元颜色集的函数，使得对于任意颜色$i$和$j$，每个颜色为$i$的字都恰好有$S(i,j)$个颜色为$j$的邻居（汉明距离为$1$），其中系数$S(i,j)$仅依赖于$i$和$j$，而不依赖于字的具体选择。所有系数$S(i,j)$构成的$k \times k$表被称为商矩阵。我们刻画了$Q_n$的次数至多为$3$的完美着色，即其商矩阵的所有特征值均不小于$n-6$，或者等价地，使得每种颜色对应于一个在$R$上可由次数不超过$3$的多项式表示的布尔函数。此外，我们刻画了$Q_n$的$(n-4)$阶相关免疫完美着色，其所有颜色对应于$(n-4)$阶相关免疫布尔函数，或者等价地，其商矩阵的所有非主（不同于$n$的）特征值均不大于$6-n$。关键词：完美着色，均衡划分，弹性函数，相关免疫函数。