Optimal redundancy of function-correcting codes

2025-02-24

AI Summary

This paper investigates the optimal redundancy of function-correcting codes—i.e., the minimal additional information required to reliably recover specific function values (e.g., Hamming weight and its distribution) under at most $t$ coordinate errors. Methodologically, it establishes a novel, deep connection between function-correcting code construction and Gray codes, integrating combinatorial coding design, extremal set theory, and asymptotic analysis. Key contributions include: (i) significantly tightening the redundancy bounds for the Hamming weight function—improving the lower bound to $4t - frac{4}{3}sqrt{6t+2} + 2$ and the upper bound to $4t - log t$; and (ii) discovering, for the first time, a phase-transition phenomenon for the Hamming weight distribution function: redundancy is exactly $2t$ when $T ge t+1$, yet achieves $4t - o(t)$ when $T = o(t)$. Collectively, these results nearly determine the optimal redundancy for the Hamming weight function.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Function-correcting codes, introduced by Lenz, Bitar, Wachter-Zeh, and Yaakobi, protect specific function values of a message rather than the entire message. A central challenge is determining the optimal redundancy -- the minimum additional information required to recover function values amid errors. This redundancy depends on both the number of correctable errors $t$ and the structure of message vectors yielding identical function values. While prior works established bounds, key questions remain, such as the optimal redundancy for functions like Hamming weight and Hamming weight distribution, along with efficient code constructions. In this paper, we make the following contributions: (1) For the Hamming weight function, we improve the lower bound on optimal redundancy from $frac{10(t-1)}{3}$ to $4t - frac{4}{3}sqrt{6t+2} + 2$. On the other hand, we provide a systematical approach to constructing explicit FCCs via a novel connection with Gray codes, which also improve the previous upper bound from $frac{4t-2}{1 - 2sqrt{log{2t}/(2t)}}$ to $4t - log{t}$. Consequently, we almost determine the optimal redundancy for Hamming weight function. (2) The Hamming weight distribution function is defined by the value of Hamming weight divided by a given positive integer $T$. Previous work established that the optimal redundancy is $2t$ when $T>2t$, while the case $T le 2t$ remained unclear. We show that the optimal redundancy remains $2t$ when $T ge t+1$. However, in the surprising regime where $T = o(t)$, we achieve near-optimal redundancy of $4t - o(t)$. Our results reveal a significant distinction in behavior of redundancy for distinct choices of $T$.

Problem

• Determine optimal redundancy for function-correcting codes
• Improve bounds for Hamming weight function redundancy
• Analyze redundancy behavior for Hamming weight distribution function

Innovation

• Improved lower bound redundancy
• Systematic FCC construction approach
• Optimal redundancy for weight distribution