On perfect 2-colorings of the q-ary n-cube

TL;DR

This paper investigates the structure of perfect 2-colorings in q-ary hypercubes, establishing inequalities relating correlation immunity and neighbor distributions, and characterizes when a coloring is perfect, also providing bounds for components and codes.

Contribution

It proves a key inequality linking correlation immunity and neighbor counts, characterizes perfect colorings via equality conditions, and offers new bounds for components and codes in q-ary hypercubes.

Findings

Established an inequality involving set density, correlation immunity, and neighbor counts.

Characterized perfect colorings as cases where the inequality becomes equality.

Provided new lower bounds for the size of components and 1-perfect codes for q>2.

Abstract

A coloring of the $q$-ary $n$-dimensional cube (hypercube) is called perfect if, for every $n$-tuple $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean-valued function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-dimensional face of the hypercube. Let $f=\chi^S$ be a characteristic function of some subset $S$ of hypercube. In the present paper it is proven the inequality $\rho(S)q({\rm cor}(f)+1)\leq \alpha(S)$, where ${\rm cor}(f)$ is the maximum degree of the correlation immunity of $f$, $\alpha(S)$ is the average number of neighbors in the set $S$ for $n$-tuples in the complement of a set $S$, and $\rho(S)=|S|/q^n$ is the density of the set $S$. Moreover, the function $f$ is a perfect coloring if and only if we obtain an equality in the above formula.Also we find new lower bound for the cardinality of components of perfect coloring and 1-perfect code in the case $q>2$. Keywords: hypercube, perfect coloring, perfect code, MDS code, bitrade, equitable partition, orthogonal array.