On the Asymptotic Rate of Optimal Codes that Correct Tandem Duplications for Nanopore Sequencing

TL;DR

This paper analyzes the asymptotic rates of optimal codes capable of correcting tandem-duplication errors in nanopore sequencing, providing exact rates or bounds depending on error regimes and parameters.

Contribution

It derives the asymptotic rate of optimal duplication-correcting codes in nanopore sequencing, addressing both unbounded and constant error regimes with new bounds and exact rates.

Findings

Exact asymptotic rate for unbounded errors when possible.

Redundancy of optimal codes is t log_q n + O(1) for constant errors when ll k.

Provides bounds on asymptotic rates depending on parameters.

Abstract

We study codes that can correct backtracking errors during nanopore sequencing. In this channel, a sequence of length $n$ over an alphabet of size $q$ is being read by a sliding window of length $\ell$, where from each window we obtain only its composition. Backtracking errors cause some windows to repeat, hence manifesting as tandem-duplication errors of length $k$ in the $\ell$-read vector of window compositions. While existing constructions for duplication-correcting codes can be straightforwardly adapted to this model, even resulting in optimal codes, their asymptotic rate is hard to find. In the regime of unbounded number of duplication errors, we either give the exact asymptotic rate of optimal codes, or bounds on it, depending on the values of $k$, $\ell$ and $q$. In the regime of a constant number of duplication errors, $t$, we find the redundancy of optimal codes to be $t\log_q n+O(1)$ when $\ell|k$, and only upper bounded by this quantity otherwise.