Nanopore sequencing, superior to other sequencing technologies for DNA storage in multiple aspects, has recently attracted considerable attention. Its high error rates, however, demand thorough research on practical and efficient coding schemes to enable accurate recovery of stored data. To this end, we consider a simplified model of a nanopore sequencer inspired by Mao \emph{et al.}, incorporating intersymbol interference and measurement noise. Essentially, our channel model passes a sliding window of length \(\ell\) over a \(q\)-ary input sequence that outputs the \textit{composition} of the enclosed \(\ell\) bits and shifts by \(\delta\) positions with each time step. In this context, the composition of a \(q\)-ary vector $\bfx$ specifies the number of occurrences in \(\bfx\) of each symbol in \(\lbrace 0,1,\ldots, q-1\rbrace\). The resulting compositions vector, termed the \emph{read vector}, may also be corrupted by \(t\) substitution errors. By employing graph-theoretic techniques, we deduce that for \(\delta=1\), at least \(\log \log n\) symbols of redundancy are required to correct a single (\(t=1\)) substitution. Finally, for \(\ell \geq 3\), we exploit some inherent characteristics of read vectors to arrive at an error-correcting code that is of optimal redundancy up to a (small) additive constant for this setting. This construction is also found to be optimal for the case of reconstruction from two noisy read vectors.

翻译：纳米孔测序在DNA存储的多个方面优于其他测序技术，近期引起了广泛关注。然而，其高错误率要求对实用且高效的编码方案进行深入研究，以实现存储数据的准确恢复。为此，我们基于Mao等人的工作，建立了一个简化的纳米孔测序通道模型，该模型包含了符号间干扰和测量噪声。本质上，我们的通道模型通过一个长度为\(\ell\)的滑动窗口对\(q\)元输入序列进行处理，输出窗口内\(\ell\)个比特的\textit{构成}，并在每次时间步后移动\(\delta\)个位置。在此背景下，\(q\)元向量\(\bfx\)的构成定义了\(\lbrace 0,1,\ldots, q-1\rbrace\)中每个符号在\(\bfx\)中出现的次数。得到的构成向量（称为\textit{读出向量}）也可能受到\(t\)个替换错误的污染。通过采用图论技术，我们推导出：当\(\delta=1\)时，至少需要\(\log \log n\)个符号的冗余来纠正单个（\(t=1\)）替换错误。最后，对于\(\ell \geq 3\)，我们利用读出向量的某些固有特性，构造了一种针对该场景冗余度最优（相差一个小常数）的错误纠正码。此外，该构造在从两个含噪读出向量进行重建的问题中也被证明是最优的。