Reconstruction of Sequences Distorted by Two Insertions

TL;DR

This paper determines the minimal redundancy of binary reconstruction codes capable of correcting two insertion errors across all numbers of noisy reads, advancing the understanding of error correction in storage systems.

Contribution

It provides the first asymptotic characterization of minimum redundancy for codes correcting exactly two insertions in binary channels for all read counts.

Findings

Minimum redundancy asymptotically determined for all N ≥ 5

Extends prior work from single errors to double insertions

Applicable to modern storage technologies like DNA storage

Abstract

Reconstruction codes are generalizations of error-correcting codes that can correct errors by a given number of noisy reads. The study of such codes was initiated by Levenshtein in 2001 and developed recently due to applications in modern storage devices such as racetrack memories and DNA storage. The central problem on this topic is to design codes with redundancy as small as possible for a given number $N$ of noisy reads. In this paper, the minimum redundancy of such codes for binary channels with exactly two insertions is determined asymptotically for all values of $N\ge 5$. Previously, such codes were studied only for channels with single edit errors or two-deletion errors.