Two-Dimensional Array Coloring With Many Colors

Given an mtimesn array and k distinct colors with 2lesmlesn and 2lesk<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 2lesklesmn/2, the maximum coloring distance is given by lfloorradic2krfloor if kleslceilm ²/2rceil, and by m+lfloor(k-lceilm ²/2rceil)/mrfloor if lceilm ²/2rceillesklesmn/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-lceilm ²/2rceil)/mrfloor if mn/2<k<mn-lfloorm ²/2rfloor, and by m+n-lceilradic2(mn-k)rceil if mn-lfloorm ²/2rfloorlesk<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 kgesmn-lceilm ²/2rceil.