The Error Probability of Maximum-Likelihood Decoding over Two Deletion Channels

Abstract: This paper studies the problem of reconstructing a word given several of its noisy copies. This setup is 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. The main focus of this paper is the case of two deletion channels and studying the error probability of the maximum-likelihood (ML) decoder under this setup. First, it is discussed how the ML decoder operates. Then, 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 error probability of the ML decoder is roughly $\frac{3q-1}{q-1}p^2$, 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. Lastly, the insertion channel is studied as well. These theoretical results are verified by corresponding simulations.