Non-Existence of Some Function-Correcting Codes With Data Protection

2026-03-01

AI Summary

This study investigates the existence of strict functional error-correcting codes—denoted as $(f\!:\!d_d,d_f)$-FCCs—which simultaneously protect both data and function values while ensuring that the function distance strictly exceeds the data distance. By constructing a distance graph among codewords and integrating tools from algebraic coding theory and combinatorial analysis, the work systematically uncovers fundamental structural limitations inherent to such codes. It is rigorously shown that classical coding schemes, including perfect codes and maximum distance separable (MDS) codes, cannot satisfy the stringent requirements of strict FCCs. Furthermore, the paper establishes several impossibility criteria that delineate essential theoretical boundaries, thereby offering critical guidance for the future design of functional error-correcting codes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In this paper, we consider the recently introduced concept of \emph{function-correcting codes (FCCs) with data protection}, which provide a certain level of error protection for the data and a higher level of protection for a desired function on the data. These codes are denoted by $(f\!:\!d_d,d_f)$-FCC, where $d_d$ is the minimum distance of the code and $d_f$ denotes the minimum distance between those codewords that correspond to different function values of a function $f:\mathbb{F}_q^k \to \mathrm{Im}(f)$, with $d_f \geq d_d$. We use a distance graph on a code based on the pairwise distances of its codewords, and show conditions under which a code cannot work as a \emph{strict} $(f\!:\!d_d,d_f)$-FCC, that is, code for which $d_f > d_d$. We then consider some well-known classes of codes, such as perfect codes and maximum distance separable (MDS) codes, and show that they cannot be used as \emph{strict} $(f\!:\!d_d,d_f)$-FCCs.

Problem

• function-correcting codes
• data protection
• minimum distance
• perfect codes
• MDS codes

Innovation

• function-correcting codes
• data protection
• distance graph
• MDS codes
• perfect codes