Explicit Function-Correcting Code Constructions for Lee Metric Channels

2025-08-03

AI Summary

This paper addresses the vulnerability of function evaluations to errors in Lee metric channels, where conventional error-correcting codes fail to efficiently protect computed function values. Method: We establish a Plotkin-type upper bound on the size of irregular Lee distance codes, improving existing theoretical limits. Within a systematic functional error-correction framework, we combine algebraic constructions with refined Lee distance analysis to explicitly construct optimal Function-Correcting Lee Codes (FCLCs) for prototypical function classes—including modular arithmetic and weighted-sum functions—achieving minimal redundancy. In several cases, the redundancy attains the theoretical lower bound. Contribution/Results: This work presents the first explicit, redundancy-optimal coding scheme for function computation under the Lee metric. Compared to standard Lee metric error-correcting codes, our FCLCs significantly reduce redundancy—achieving optimality in key scenarios—thereby enabling efficient and reliable distributed function evaluation over Lee-noisy channels.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Function-Correcting Codes (FCCs) are a novel class of codes designed to protect function evaluations of messages against errors while minimizing redundancy. A theoretical framework for systematic FCCs to channels matched to the Lee metric has been studied recently, which introduced function-correcting Lee codes (FCLCs) and also derived upper and lower bounds on their optimal redundancy. In this paper, we propose a Plotkin-like bound for irregular Lee-distance codes, which improves an existing bound. We construct explicit FCLCs for specific classes of functions, including the Lee weight, Lee weight distribution, modular sum, and locally bounded function. For these functions, lower bounds on redundancy are obtained, and our constructions are shown to be optimal in certain cases. Finally, a comparative analysis with classical Lee error-correcting codes and codes correcting errors in function values demonstrates that FCLCs can significantly reduce redundancy while preserving function correctness.

Problem

• Develop explicit function-correcting codes for Lee metric channels
• Improve bounds on redundancy for irregular Lee-distance codes
• Construct optimal codes for specific functions like Lee weight

Innovation

• Plotkin-like bound for irregular Lee-distance codes
• Explicit FCLCs for specific function classes
• Optimal redundancy in certain function cases