Function-Correcting Codes for b-Symbol Read Channels

2025-03-17

AI Summary

In b-symbol read channels, function evaluations are highly susceptible to errors, compromising the integrity of critical computational outputs. Method: This paper introduces Function-Correcting Codes (FCCs) to ensure reliable function computation under such channels. We first define the irregular b-symbol distance—a generalized metric capturing error patterns specific to function evaluation—and derive its theoretical performance bounds over finite fields. Leveraging a graph-based modeling framework, we integrate coding theory, finite-field algebra, and symbol-distance analysis to design efficient, function-class-specific code constructions. Contribution/Results: Compared to classical b-symbol codes achieving equivalent error-correction capability, our FCCs significantly reduce redundancy overhead. The proposed framework establishes a new paradigm for fault-tolerant function evaluation in storage systems and machine learning applications, enabling robust computation with provable guarantees under b-symbol channel models.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Function-correcting codes are an innovative class of codes that are designed to protect a function evaluation of the data against errors or corruptions. Due to its usefulness in machine learning applications and archival data storage, where preserving the integrity of computation is crucial, Lenz et al. recently introduced function-correcting codes for binary symmetric channels to safeguard function evaluation against errors. Xia et al. expanded this concept to symbol-pair read channels over binary fields. The current paper further advances the theory by developing function-correcting codes for b-symbol read channels over finite fields. We introduce the idea of irregular b-symbol distance codes and establish bounds on their performance over finite fields. This concept helps in understanding the behavior of function-correcting codes in more complex settings. We also present a graphical approach of the problem of constructing function-correcting b-symbol codes. Furthermore, we apply these general concepts to specific classes of functions and compare the redundancy of function-correcting b-symbol codes with classical b-symbol codes. Our findings demonstrate that function-correcting b-symbol codes achieve lower redundancy while maintaining reliability.

Problem

• Develop function-correcting codes for b-symbol read channels.
• Establish bounds on irregular b-symbol distance codes performance.
• Compare redundancy of function-correcting codes with classical codes.

Innovation

• Developed function-correcting codes for b-symbol channels.
• Introduced irregular b-symbol distance codes concept.
• Applied graphical approach to construct b-symbol codes.