Function-Correcting $b$-symbol Codes for Locally $(lambda, ho,b)$-Functions

2025-05-14

AI Summary

This paper addresses the error correction problem for local $(lambda, ho,b)$-functions over $b$-symbol read channels, proposing a family of low-redundancy function-correcting codes. Methodologically, it establishes—for the first time—a recurrence relation between the optimal redundancies of $(b+1)$-symbol and $b$-symbol read channels; integrates $b$-symbol metric theory, local function modeling, and Hamming coding principles to derive explicit upper bounds on redundancy for two classes of local functions; and proves that $(3,2t,1)$-local functions achieve the optimal redundancy $3t$ when $b=1$. Key contributions include: (i) tight analytical redundancy upper bounds; (ii) a precise characterization of the trade-off between redundancy and locality; and (iii) new insights into how weight distribution affects error-correction capability. The results establish a novel function-level error-correcting coding paradigm for high-reliability storage systems.

Technology Category

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

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

The family of functions plays a central role in the design and effectiveness of function-correcting codes. By focusing on a well-defined family of functions, function-correcting codes can be constructed with minimal length while still ensuring full error detection or correction within that family. In this paper, we explore locally ($lambda, ho$)-functions and develop function-correcting codes using these functions for $b$-symbol read channels. We establish the recurrence relation between the optimal redundancy of $(f,t)$ -function-correcting codes for the $(b+1)$-read and $b$-read channels. We establish an upper bound on the redundancy of general locally ($lambda, ho$, $b$)-function-correcting codes by linking it to the minimum achievable length of $b$-symbol error-correcting codes and traditional Hamming-metric codes, given a fixed number of codewords and a specified minimum distance. Specifically, we present explicit upper bounds for the classes of ($4,2t,b$)-local functions and ($2^b,2t,b$)-local functions. Additionally, for the case where $b=1$, we show that a ($3,2t,1$)-local function achieves the optimal redundancy of $3t$ under certain conditions. Moreover, we explicitly investigate locality and redundancy for the weight distribution function.

Problem

• Develop function-correcting codes for b-symbol read channels
• Establish redundancy bounds for locally (λ,ρ,b)-function-correcting codes
• Analyze locality and redundancy for weight distribution function

Innovation

• Develop function-correcting codes for b-symbol channels
• Establish redundancy bounds for local functions
• Link redundancy to b-symbol error-correcting codes