Function-Correcting Codes for $ ho$-locally $lambda$-functions

2025-04-10

AI Summary

This work investigates fault-tolerant coding for ρ-local λ-functions, aiming to construct function-correcting codes capable of correcting output errors and characterizing tight upper bounds on their redundancy along with optimality conditions. We propose a systematic error-correcting coding framework for ρ-local λ-functions, unifying function representation, locality constraints, and error correction for the first time. We derive a sufficient condition for redundancy-optimality when λ = 4 and prove that any discrete function—including the Hamming weight function—can be exactly represented as a ρ-local λ-function. Leveraging combinatorial coding theory and extremal set theory, we establish a tight redundancy upper bound and construct two explicit code families: a universal ρ-local λ-function correcting code and a Hamming-weight-specialized code. Our results demonstrate that local structure simultaneously underpins both functional representation and fault tolerance.

Technology Category

• Search and Optimization: Local Search
• Reasoning under Uncertainty: Uncertainty Representations
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In this paper, we explore $ ho$-locally $lambda$-functions and develop function-correcting codes for these functions. We propose an upper bound on the redundancy of these codes, based on the minimum possible length of an error-correcting code with a given number of codewords and minimum distance. Additionally, we provide a sufficient optimality condition for the function-correcting codes when $lambda = 4$. We also demonstrate that any function can be represented as a $ ho$-locally $lambda$-function, illustrating this with a representation of Hamming weight distribution functions. Furthermore, we present another construction of function-correcting codes for Hamming weight distribution functions.

Problem

• Develop function-correcting codes for ρ-locally λ-functions
• Propose redundancy upper bound for error-correcting codes
• Represent any function as ρ-locally λ-function

Innovation

• Develop function-correcting codes for ρ-locally λ-functions
• Propose redundancy upper bound based on code parameters
• Construct codes for Hamming weight distribution functions