Function-Correcting Codes for Insertion-Deletion Channel

2025-12-08

AI Summary

This work addresses the fundamental problem of redundancy optimization for insertion-deletion (insdel) channels. We introduce, for the first time, the functional error-correcting codes (FECCs) framework into the insdel setting, proposing a unified functional protection mechanism capable of correcting insertions, deletions, and their combinations. By defining an irregular insdel distance and an insdel distance matrix, we establish the equivalence among three classes of FECCs. Leveraging Gilbert–Varshamov– and Plotkin-type bounds—combined with modeling via classic functional constraints such as VT syndromes and run-length counts—we derive tight upper and lower bounds on optimal redundancy for several key functions over insdel channels. The resulting length–redundancy theoretical framework substantially reduces encoding overhead, enabling low-redundancy, high-reliability functional message protection—a novel paradigm for DNA-based data storage and robust machine learning systems.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In coding theory, handling errors that occur when symbols are inserted or deleted from a transmitted message is a long-standing challenge. Optimising redundancy for insertion and deletion channels remains a key open problem with significant importance for applications in DNA data storage and document exchange. Recently, a coding framework known as function-correcting codes has been proposed to address the challenge of minimising redundancy while preserving specific functions of the message. This framework has gained attention due to its potential applications in machine learning systems and long-term archival data storage. Motivated by the problem of redundancy optimisation for insertion and deletion channels, we propose a new framework called function-correcting codes for insdel channels. In this paper, we introduce the notions of function-correcting insertion codes, function-correcting deletion codes, and function-correcting insdel codes, and we show that these three formulations are equivalent. We then define insdel distance matrices and irregular insdel-distance codes, and derive lower and upper bounds on the optimal redundancy achievable by function-correcting codes for insdel channels. In addition, we establish Gilbert-Varshamov and Plotkin-like bounds on the length of irregular insdel-distance codes. Using the relation between optimal redundancy and the length of such codes, we obtain a simplified lower bound on optimal redundancy. Finally, we derive bounds on the optimal redundancy of function-correcting insdel codes for several classes of functions, including locally bounded functions, VT syndrome functions, the number-of-runs function, and the maximum-run-length function.

Problem

• Optimizes redundancy for insertion-deletion channels in coding theory.
• Proposes function-correcting codes to preserve message functions with minimal redundancy.
• Derives bounds on redundancy for specific functions in insdel channels.

Innovation

• Function-correcting codes for insertion-deletion channels
• Equivalent formulations for insertion, deletion, and insdel codes
• Bounds on optimal redundancy for various function classes