Optimal redundancy of function-correcting codes

TL;DR

This paper investigates the minimal additional information needed to protect specific function values of messages against errors, focusing on Hamming weight and weight distribution functions, providing improved bounds and explicit constructions.

Contribution

The paper improves bounds on optimal redundancy for function-correcting codes and introduces a novel Gray code-based construction, nearly determining the redundancy for Hamming weight functions.

Findings

Improved lower bound on redundancy for Hamming weight function.

Explicit Gray code-based construction for FCCs with better upper bounds.

Redundancy behavior varies significantly with the parameter T in weight distribution functions.

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 $\frac{4t-2}{1 - 2\sqrt{\log{2t}/(2t)}}$ 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 $T$.