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This paper tackles two problems that fall under the study of coding for insertions and deletions. These problems are motivated by several applications, among them is reconstructing strands in DNA-based storage systems. Under this paradigm, a word is transmitted over some fixed number of identical independent channels and the goal of the decoder is to output the transmitted word or some close approximation of it. The first part of this paper studies the deletion channel that deletes a symbol with some fixed probability p, while focusing on two instances of this channel. Since operating the maximum likelihood (ML) decoder in this case is computationally unfeasible, we study a slightly degraded version of this decoder for two channels and study its expected normalized distance. We observe that the dominant error patterns are deletions in the same run or errors resulting from alternating sequences. Based on these observations, it is derived that the expected normalized distance of the degraded ML decoder is roughly 3q−1 q−1 p 2 , when the transmitted word is any q-ary sequence and p is the channel's deletion probability. We also study the cases when the transmitted word belongs to the Varshamov Tenengolts (VT) code or the shifted VT code. Additionally, the insertion channel is studied as well as the case of two insertion channels. These theoretical results are verified by corresponding simulations. The second part of the paper studies optimal decoding for a special case of the deletion channel, referred by the k-deletion channel, which deletes exactly k symbols of the transmitted word uniformly at random. In this part, the goal is to understand how an optimal decoder operates in order to minimize the expected normalized distance. A full characterization of an efficient optimal decoder for this setup, reffered to as the maximum likelihood* (ML*) decoder, is given for a channel that deletes one or two symbols. For k = 1 it is shown that when the code is the entire space, the decoder is the lazy decoder which simply returns the channel output. Similarly, for k = 2 it is shown that the decoder acts as the lazy decoder in almost all cases and when the longest run is significantly long (roughly (2 − √ 2)n when n is the word length), it prolongs the longest run by one symbol.

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