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.

中文翻译：

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两个删除信道上最大似然解码的错误概率

本文研究了在给定几个噪声副本的情况下重建单词的问题。这种设置是由几个应用程序驱动的，其中包括在基于 DNA 的存储系统中重建链。在这种范式下，一个字通过一些固定数量的相同独立通道传输，解码器的目标是输出传输的字或一些近似值。本文的主要重点是两个删除通道的情况，并研究了这种设置下最大似然 (ML) 解码器的错误概率。首先，讨论ML解码器如何操作。然后，我们观察到主要错误模式是同一运行中的删除或由交替序列引起的错误。基于这些观察，推导出ML解码器的错误概率大致为$\frac{3q-1}{q-1}p^2$，当传输的词是任何 $q$-ary 序列并且 $p$ 是通道的删除概率时。我们还研究了传输的字属于 Varshamov Tenengolts (VT) 码或移位 VT 码的情况。最后，还研究了插入通道。这些理论结果通过相应的模拟得到验证。