Two-Dimensional Array Coloring With Many Colors

Given an m times n array and k distinct colors with 2les m les n and 2les k < mn , we consider the problem of marking each location in the array using one of the k given colors such that any two locations in the array marked by the same color are separated as much as possible. This problem is related to two-dimensional (2-D) interleaving schemes for correcting cluster errors where the goal is to rearrange the codeword symbols so that an arbitrarily shaped error cluster of size t can be corrected for the largest possible value of t . In a recent paper, the authors have shown that, for the case 2les k les mn /2, the maximum coloring distance is given by lfloorradic2 k rfloor if k leslceil m 2 /2rceil, and by m +lfloor( k -lceil m 2 /2rceil)/ m rfloor if lceil m 2 /2rceilles k les mn /2. In this work, we extend these results to the case mn /2< k < mn . We show that in such cases, the maximum coloring distance is given by to m +lfloor( k -lceil m 2 /2rceil)/ m rfloor if mn /2< k < mn -lfloor m 2 /2rfloor, and by m + n -lceilradic2( mn - k )rceil if mn -lfloor m 2 /2rfloorles k < mn . In particular, we generalize the partial sphere packing argument to derive the new bound and consequently propose a new type of construction achieving optimal coloring for the case k ges mn -lceil m 2 /2rceil.