Repeated deletion channels

We consider the question of the capacity of the deletion channel when multiple copies of the codeword are sent independently through the channel. This type of coding problem is straightforward for standard erasure and error channels, but is more challenging in the deletion channel setting. Our results show that when the same codeword is sent k times through the channel, each time with independent deletions with probability d, the optimal communication rate under a random codebook is 1 - αH(d^k) + O(d^k) for an explicitly given constant α.