New bounds on a hypercube coloring problem and linear codes

Abstract

In studying the scalability of optical networks, one problem arising involves coloring the vertices of the n-dimensional hypercube 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 /spl chi//sub k~/(n), the minimum number of colors needed, appears to be a difficult problem. We improve the known lower and upper bounds of /spl chi//sub k~/(n) and indicate the connection of this colouring problem to linear codes.