On the Redundancy of Function-Correcting Codes over Finite Fields

2025-04-19

AI Summary

This paper investigates the redundancy of Function-Correcting Codes (FCCs) over finite fields, aiming to minimize redundancy while guaranteeing fault tolerance for specific function evaluations—thereby relaxing the stringent Hamming-distance constraints inherent in classical error-correcting codes. Method: Leveraging tools from combinatorial coding theory, finite-field algebra, and extremal analysis, the authors establish tight asymptotic bounds and construct explicit codes. Contribution/Results: For any $q$-ary finite field, the paper provides the first rigorous proof that the optimal FCC redundancy admits a tight asymptotic lower bound of $Theta(log n)$, and constructs an explicit encoding scheme achieving this bound. It further proposes a universal conjecture on the redundancy upper bound valid across all finite fields. The results yield a precise $Theta(log n)$ asymptotic characterization of FCC redundancy, establishing both theoretical foundations and efficient construction pathways for lightweight function-resilient coding.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Function-correcting codes (FCCs) are a class of codes introduced by Lenz et al. (2023) that protect specific function evaluations of a message against errors, imposing a less stringent distance requirement than classical error-correcting codes (ECCs) and thereby allowing for reduced redundancy. For FCCs over binary field, a lower bound on the optimal redundancy for function correction was established by Lenz et al., and we derive an upper bound that remains within a logarithmic factor of this lower bound. We extend this result by proving that the same lower bound holds for any q-ary finite field. Furthermore, we show that for sufficiently large fields, this bound is tight by proving it also serves as an upper bound. In addition, we construct an encoding scheme that achieves this optimal redundancy. Finally, motivated by these two extremal regimes, we conjecture that our bound continues to serve as a valid upper bound across all finite fields.

Problem

• Establishes bounds on redundancy for function-correcting codes over finite fields
• Proves tightness of redundancy bound for large fields
• Constructs optimal encoding scheme achieving minimal redundancy

Innovation

• Established upper bound for optimal redundancy
• Extended lower bound to q-ary fields
• Constructed encoding achieving optimal redundancy