On a hypercube coloring problem

Let χ k ¯ ( n ) denote the minimum number of colors necessary to color the n-dimensional hypercube so that no two vertices that are at a distance at most k from each other get the same color. In other words, this is the smallest number of binary codes with minimum distance k + 1 that form a partition of the n-dimensional binary Hamming space. It is shown that χ 2 ¯ ( n ) ∼ n and χ 3 ¯ ( n ) ∼ 2 n as n tends to infinity.