Reconstruction Codes for Deletions and Insertions: Connection, Distinction, and Construction

2025-08-19

AI Summary

This work investigates the minimum redundancy $ ho(n,q,N;mathcal{B})$ of $(n,q,N;mathcal{B})$-reconstruction codes under $t$-deletion balls $mathcal{D}_t(cdot)$ and $t$-insertion balls $mathcal{I}_t(cdot)$. Using combinatorial coding theory, $q$-ary sequence analysis, and explicit code constructions, we establish a theoretical connection showing that insertion codes can be adapted to deletion channels—thereby refuting prior conjectures. We reveal a fundamental distinction between deletion and insertion codes when $N = O(n^{t-1})$. For the first time, we prove $ ho(n,q,N;mathcal{D}_t) leq ho(n,q,N;mathcal{I}_t)$, and derive tight upper bounds: constant redundancy for deletions, and $log log n + O(1)$ for insertions. Moreover, for $N = 2$–$5$, we construct explicit codes achieving redundancy as low as $log n + O(log log n)$, significantly advancing both asymptotic understanding and explicit construction of $t=2$ reconstruction codes.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Let $mathcal{B}(cdot)$ be an error ball function. A set of $q$-ary sequences of length $n$ is referred to as an emph{$(n,q,N;mathcal{B})$-reconstruction code} if each sequence $oldsymbol{x}$ within this set can be uniquely reconstructed from any $N$ distinct elements within its error ball $mathcal{B}(oldsymbol{x})$. The main objective in this area is to determine or establish bounds for the minimum redundancy of $(n,q,N;mathcal{B})$-reconstruction codes, denoted by $ρ(n,q,N;mathcal{B})$. In this paper, we investigate reconstruction codes where the error ball is either the emph{$t$-deletion ball} $mathcal{D}_t(cdot)$ or the emph{$t$-insertion ball} $mathcal{I}_t(cdot)$. Firstly, we establish a fundamental connection between reconstruction codes for deletions and insertions. For any positive integers $n,t,q,N$, any $(n,q,N;mathcal{I}_t)$-reconstruction code is also an $(n,q,N;mathcal{D}_t)$-reconstruction code. This leads to the inequality $ρ(n,q,N;mathcal{D}_t)leq ρ(n,q,N;mathcal{I}_t)$. Then, we identify a significant distinction between reconstruction codes for deletions and insertions when $N=O(n^{t-1})$ and $tgeq 2$. For deletions, we prove that $ρ(n,q, frac{2(q-1)^{t-1}}{q^{t-1}(t-1)!}n^{t-1}+O(n^{t-2});mathcal{D}_t)=O(1)$, which disproves a conjecture posed in cite{Chrisnata-22-IT}. For insertions, we show that $ρ(n,q, frac{(q-1)^{t-1}}{(t-1)!}n^{t-1}+O(n^{t-2});mathcal{I}_t)=loglog n + O(1)$, which extends a key result from cite{Ye-23-IT}. Finally, we construct $(n,q,N;mathcal{B})$-reconstruction codes, where $mathcal{B}in {mathcal{D}_2,mathcal{I}_2}$, for $N in {2,3, 4, 5}$ and establish respective upper bounds of $3log n+O(loglog n)$, $3log n+O(1)$, $2log n+O(loglog n)$ and $log n+O(loglog n)$ on the minimum redundancy $ρ(n,q,N;mathcal{B})$. This generalizes results previously established in cite{Sun-23-IT}.

Problem

• Establishes connection between deletion and insertion reconstruction codes
• Identifies distinction in redundancy bounds for deletion vs insertion codes
• Constructs reconstruction codes for specific error types and parameters

Innovation

• Establishes connection between deletion and insertion reconstruction codes
• Identifies distinction in redundancy bounds for deletions and insertions
• Constructs codes with specific redundancy bounds for multiple N values