Function-Correcting Codes With Data Protection

TL;DR

This paper introduces a comprehensive framework for function-correcting codes that simultaneously protect data and function values, including explicit constructions, bounds, and analysis of linear codes.

Contribution

It proposes a general approach and a two-step construction for FCCs with data protection, extending classical bounds and analyzing linear FCCs.

Findings

Data protection can be added to existing FCCs without increasing redundancy.

Explicit constructions for specific functions like Hamming weight are provided.

Perfect and MDS codes cannot enhance function value protection beyond data protection.

Abstract

Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. We first introduce a more general approach and a framework for function-correcting codes that incorporates data protection along with protection of function values. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph called minimum-distance graph to a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. To the best of our knowledge, this is the first instance of FCCs with a linear structure. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.