New bounds on a hypercube coloring problem

In studying the scalability of optical networks, one problem which arises involves coloring the vertices of the n-cube with as few colors as possible such that any two vertices whose Hamming distance is at most k are colored differently. Determining the exact value of χ k ̄ (n) , the minimum number of colors needed, appears to be a difficult problem. In this note, we improve the known lower and upper bounds of χ k ̄ (n) and indicate the connection of this coloring problem to linear codes.