Achievable rates for channels with deletions and insertions

Consider a binary channel with deletions and insertions, where each input bit is transformed in one of the following ways: it is deleted with probability d, or an extra bit added after it with probability i, or it is transmitted unmodified with probability 1 - d - i. We obtain a lower bound on the capacity of this channel. The transformation of the input sequence by the channel may be viewed in terms of runs as follows: some runs of the input sequence get shorter/longer, some runs get deleted, and some new runs are added. The capacity is difficult to compute mainly due to the last two phenomena: deleted runs, and new inserted runs. We consider a decoder which first decodes the positions of the deleted and inserted runs, and then the transmitted codeword. Analyzing the performance of such a decoder leads to a computable lower bound on the capacity.