Optimal redundancy of function-correcting codes
Gennian Ge, Zixiang Xu, Xiande Zhang, Yijun Zhang
TL;DR
This work advances the theory of function-correcting codes (FCCs) by precisely characterizing the redundancy required to protect key function values of binary messages under errors. It develops the distance-demand/irregular-distance code framework to relate FCC redundancy to combinatorial packing via the distance-demand matrix, and then derives tight results for two principal function classes: the Hamming weight and the Hamming weight distribution. For Hamming weight, the authors fuse a novel Gray-code–based construction with refined counting arguments to achieve near-optimal redundancy, lowering the bound to $r_{ ext{wt}}(k,t) \ge 4t - \frac{4}{3}\sqrt{6t+2} + 2$ and tightening the upper bound to $r_{ ext{wt}}(k,t) \le 4t - \log t$. For the weight distribution, they establish distinct regimes: $r_{\Delta_T}(k,t)=2t$ when $T \ge t+1$ and, when $T = o(t)$, a near-optimal $4t - o(t)$ bound, with detailed constructions and bounds depending on $T$ and the quotient $\frac{2t+1}{T}$. The paper further provides a general construction pathway via Gray codes and specific linear codes (e.g., Simplex and Belov-type codes) to realize explicit FCCs with improved redundancy bounds. Overall, the results reveal two different redundancy behaviors under different function definitions and offer practically relevant, explicit code constructions that substantially narrow the gap toward optimal redundancy. The findings have potential implications for storage and communication systems where only particular function values need protection, and they point to extensions to symbol-pair FCCs and other function families.
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 $T$.