Sequence Reconstruction for Single-Deletion Single-Substitution Channel

2025-01-07

AI Summary

This paper investigates the unique reconstructibility of $q$-ary sequences over a single-deletion–single-substitution hybrid channel: given a codebook with minimum Hamming distance at least 2, it determines the minimum number of distorted outputs required to uniquely recover the original sequence. We introduce an error-ball intersection modeling framework and derive, for the first time, a tight upper bound on the size of the intersection of error balls centered at any two distinct codewords: $2q^{n-3} - 2q - 2 - delta_{q,2}$. Tightness is established via explicit construction of codeword pairs achieving this bound. This result characterizes the fundamental theoretical limit on the minimum number of outputs needed for sequence reconstruction under this channel model. It significantly advances the capacity analysis of codes resilient to combined deletion and substitution errors and provides critical theoretical foundations for designing robust, error-correcting codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The central problem in sequence reconstruction is to find the minimum number of distinct channel outputs required to uniquely reconstruct the transmitted sequence. According to Levenshtein's work in 2001, this number is determined by the maximum size of the intersection between the error balls of any two distinct input sequences of the channel. In this work, we study the sequence reconstruction problem for single-deletion single-substitution channel, assuming that the transmitted sequence belongs to a $q$-ary code with minimum Hamming distance at least $2$, where $qgeq 2$ is any fixed integer. Specifically, we prove that for any two $q$-ary sequences with Hamming distance $dgeq 2$, the size of the intersection of their error balls is upper bounded by $2qn-3q-2-delta_{q,2}$, where $delta_{i,j}$ is the Kronecker delta. We also prove the tightness of this bound by constructing two sequences the intersection size of whose error balls achieves this bound.

Problem

• Error Correction
• Transmission Channel
• Coding Theory

Innovation

• Hamming Distance
• Error Correction
• Information Theory