On perfect 2-colorings of the -ary -cube

A coloring of a 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 value 1 the same number of times for each m -dimensional face of the hypercube. Let f = χ S be a characteristic function of a subset S of hypercube. In the present paper we prove the inequality ρ ( S ) q ( cor ( f ) + 1 ) ≤ α ( S ) , where cor ( f ) is the maximum degree of the correlation immunity of f , α ( S ) is the average number of neighbors in the set S for n -tuples in the complement of a set S , and ρ ( S ) = | S | / q n is the density of the set S . Moreover, the function f is a perfect coloring if and only if we have an equality in the formula above. Also, we find a new lower bound for the cardinality of components of a perfect coloring and a 1-perfect code in the case q > 2 .