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Nanopore sequencing, superior to other sequencing technologies for DNAstorage in multiple aspects, has recently attracted considerable attention. Itshigh error rates, however, demand thorough research on practical and efficientcoding schemes to enable accurate recovery of stored data. To this end, weconsider a simplified model of a nanopore sequencer inspired by Mao \emph{etal.}, incorporating intersymbol interference and measurement noise.Essentially, our channel model passes a sliding window of length $\ell$ overa $q$-ary input sequence that outputs the \textit{composition} of theenclosed $\ell$ bits and shifts by $\delta$ positions with each time step.In this context, the composition of a $q$-ary vector $\bfx$ specifies thenumber 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 employinggraph-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 inherentcharacteristics of read vectors to arrive at an error-correcting code that isof optimal redundancy up to a (small) additive constant for this setting. Thisconstruction is also found to be optimal for the case of reconstruction fromtwo noisy read vectors.

Error-Correcting Codes for Nanopore Sequencing