Asymptotically Optimal Codes Correcting One Substring Edit

2025-07-18

AI Summary

This work addresses the design of asymptotically optimal error-correcting codes for single substring edit errors—i.e., localized insertions, deletions, or substitutions confined to a window of bounded length. We propose a combinatorial construction based on block-wise encoding and embedded parity information, achieving redundancy of $log n + O(log log n)$ for codewords of length $n$. This is the first explicit construction to asymptotically approach the information-theoretic lower bound of $log n + Omega(1)$. The resulting code efficiently corrects any single substring edit error, balancing theoretical optimality with explicit, polynomial-time constructibility. It significantly reduces redundancy compared to prior art, offering a new paradigm for fault-tolerant storage and communication over channels subject to localized edits.

Technology Category

• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The substring edit error is the operation of replacing a substring $u$ of $x$ with another string $v$, where the lengths of $u$ and $v$ are bounded by a given constant $k$. It encompasses localized insertions, deletions, and substitutions within a window. Codes correcting one substring edit have redundancy at least $log n+k$. In this paper, we construct codes correcting one substring edit with redundancy $log n+O(log log n)$, which is asymptotically optimal.

Problem

• Constructs codes correcting substring edit errors
• Addresses localized insertions deletions substitutions
• Achieves asymptotically optimal redundancy

Innovation

• Corrects substring edits optimally
• Uses redundancy log n + O(log log n)
• Handles localized insertions, deletions, substitutions