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

TL;DR

We address the problem of classifying perfect colorings of the hypercube $Q_n$ in two regimes: (i) colorings of degree at most $3$, equivalently quotient matrices with smallest eigenvalue at least $n-6$, and (ii) $(n-4)$-correlation-immune colorings, where all non-main eigenvalues are not greater than $6-n$. The authors connect low-degree colorings to $(n-4)$-resilient Boolean functions, leveraging Rasoolzadeh's complete classification and an algorithm that exhausts colorings on $Q_{10}$ to show that for $n>10$ all colorings have nonessential arguments, resulting in a finite, constructible set up to equivalence. They also classify $(n-4)$-correlation-immune colorings for $n=5$–$9$, detailing quotient matrices, counts, and notable structures such as multifold $1$-perfect codes in $Q_7$ and special two-eigenvalue colorings. Overall, the work yields a finite catalog of low-degree colorings and explicit constructions, strengthening the links between equitable partitions, resilient/correlation-immune Boolean functions, and the spectral theory of $Q_n$.

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.

Figures (2)

• Figure 2: Four perfect $3$-colorings $f_{0,0}$, $f_{0,1}$, $f_{1,0}$, $f_{1,1}$, of $Q_4$ with quotient matrix $[[1,1,2],[1,1,2],[1,1,2]]$
• Figure 3: The perfect $4$-colorings of $Q_4$ with quotient matrix $[[1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1]]$

Theorems & Definitions (31)

• Lemma 1
• Lemma 2: folklore, see e.g. Kro:OA13
• Lemma 3
• Corollary 1
• proof
• Corollary 2
• proof
• Lemma 4: see e.g. Godsil93
• Corollary 3
• proof