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

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.