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

TL;DR

This paper investigates the theoretical limitations of function-correcting codes with data protection, proving that certain well-known code classes cannot achieve the desired function-specific error protection levels.

Contribution

It establishes non-existence results for strict function-correcting codes within classes like perfect and MDS codes, clarifying their limitations.

Findings

Perfect codes cannot serve as strict FCCs.

MDS codes are not suitable as strict FCCs.

Conditions for non-existence of certain FCCs are derived.

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.