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

Charul Rajput, B. Sundar Rajan, Ragnar Freij-Hollanti, Camilla Hollanti

TL;DR

Some well-known classes of codes, such as perfect codes and maximum distance separable (MDS) codes, are considered, and it is shown that they cannot be used as \emph{strict} $(f\!:\!d_d,d_f)$-FCCs.

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.

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

TL;DR

Some well-known classes of codes, such as perfect codes and maximum distance separable (MDS) codes, are considered, and it is shown that they cannot be used as \emph{strict} -FCCs.

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 -FCC, where is the minimum distance of the code and denotes the minimum distance between those codewords that correspond to different function values of a function , with . 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} -FCC, that is, code for which . 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} -FCCs.
Paper Structure (11 sections, 11 theorems, 24 equations, 1 figure)

This paper contains 11 sections, 11 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution and paper organization
  3. Preliminaries
  4. Function-correcting codes with Data Protection
  5. Non-existence of strict $(f\!:\!d_d,d_f)$-FCC
  6. Minimum-Distance Graphs of Perfect Codes and MDS Codes
  7. Perfect Codes
  8. MDS Codes
  9. Minimum-distance graph of linear codes
  10. $\alpha$-distance graph of some other classes of codes
  11. Conclusion

Key Result

Theorem 1

Let $\mathcal{C}$ be an $(n,q^k,d)$ code. If the $\alpha$-distance graph $G_{\alpha}(\mathcal{C})$ is a connected graph, then $\mathcal{C}$ cannot be an $(f\!:d,d_f)$-FCC for any $f:\mathbb{F}_q^k\to \text{Im}(f)$ with $|\text{Im}(f)| \geq 2$ and $d_f>\alpha$.

Figures (1)

  • Figure :

Theorems & Definitions (23)

  • Definition 1: Covering radius
  • Definition 2: Function-Correcting Codes
  • Definition 3
  • Example 1
  • Definition 4: Optimal redundancy
  • Definition 5: $\alpha$-distance graph
  • Example 2
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 13 more