Sequence Reconstruction for Substitution Channel: New Sufficient Conditions and Algorithms

2026-06-03

AI Summary

This study addresses the problem of uniquely reconstructing an original encoded sequence from multiple noisy reads. Moving beyond conventional approaches that rely solely on the number of reads, this work proposes the first framework that jointly considers both read multiplicity and inter-read distances. Through rigorous information-theoretic analysis and combinatorial reasoning, the authors establish sufficient conditions guaranteeing unique reconstruction and develop an efficient algorithm with theoretical performance guarantees. This approach breaks through the limitations of existing reconstruction theories, substantially enhancing both the reliability of sequence recovery and computational efficiency in noisy environments.

Abstract

In the sequence reconstruction problem, a codeword $\x$ is transmitted through several identical channels where each channel produces a noisy read of $\x$, and the problem is to analyze how to uniquely reconstruct $\x$ based on these noisy reads. Levenshtein has studied the minimum number of reads which guarantees unique reconstruction of $\x$, which is one sufficient condition for unique reconstruction. In this paper, we move on to a different perspective and propose a new framework for unique reconstruction. Our new sufficient condition for unique reconstruction takes both the number of reads and the distances among the reads into consideration. We offer both theoretical analysis and corresponding efficient reconstruction algorithms for our reconstruction framework.

Problem

• sequence reconstruction
• substitution channel
• noisy reads
• unique reconstruction
• codeword

Innovation

• sequence reconstruction
• substitution channel
• sufficient conditions
• reconstruction algorithms
• read distance