Coding for Sequence Reconstruction for Single Edits

TL;DR

This paper studies coding strategies for reconstructing sequences from multiple noisy reads with single edit errors, showing how redundancy can be minimized as the number of reads increases, relevant for modern storage systems.

Contribution

It introduces reconstruction codes for fixed numbers of noisy reads with single edit errors, and characterizes how redundancy decreases with more reads, achieving near-optimal efficiency.

Findings

Redundancy reduces from log n to constant as reads increase.

Reconstruction codes are within one bit of optimal redundancy.

Designed codes work for all fixed numbers of noisy reads.

Abstract

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. The common setup assumes the codebook to be the entire space and the problem is to determine the minimum number of distinct reads that is required to reconstruct the transmitted codeword. Motivated by modern storage devices, we study a variant of the problem where the number of noisy reads $N$ is fixed. Specifically, we design reconstruction codes that reconstruct a codeword from $N$ distinct noisy reads. We focus on channels that introduce single edit error (i.e. a single substitution, insertion, or deletion) and their variants, and design reconstruction codes for all values of $N$. In particular, for the case of a single edit, we show that as the number of noisy reads increases, the number of redundant bits required can be gracefully reduced from $\log n+O(1)$ to $\log \log n+O(1)$, and then to $O(1)$, where $n$ denotes the length of a codeword. We also show that the redundancy of certain reconstruction codes is within one bit of optimality.