Function-Correcting Codes With Data Protection

TL;DR

This work generalizes function-correcting codes (FCCs) to jointly protect data and function values, introducing a two-step construction that combines an error-correcting code with an FCC built on its codewords. The total redundancy becomes $r_s = n - k + r'$, and the authors develop coded distance matrices (CDRM/CFDM) to analyze function-distance requirements, linking FCCs to irregular-distance codes. They prove that perfect and MDS codes cannot provide extra protection beyond data and present explicit FCCs for locally bounded, locally binary, and Hamming weight functions, including linear FCCs via coset-code constructions. Moreover, the paper extends classical Plotkin and Hamming bounds to the FCC setting, offering generalized bounds that depend on the image size and preimage structure of the function. Collectively, these contributions enable design of FCCs with data protection, broadening applicability to networks and data storage while clarifying fundamental limits and linear-structure possibilities.

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.

Figures (6)

• Figure 1: General setup of function-correcting codes. The transmitter has a message vector ${u}$, and the receiver has a special interest in a function $f({u})$ of this message. The transmitter encodes the message ${u}$ to $c({u})$ and sends it over the erroneous channel. The receiver receives the vector $c({u}) + {e}$, and can decode the message and the function value with different levels of error protection guarantees based on the importance of the function $f$ for the receiver.
• Figure : (a) Wired network
• Figure :
• Figure : (a) Wired network
• Figure : (b) Wireless network

Theorems & Definitions (110)

• Definition 1: Function-Correcting Codes
• Definition 2: Distance Requirement Matrix (DRM)
• Example 1
• Definition 3: Irregular-distance code or $\mathcal{D}$-code
• Example 2
• Lemma 1
• Definition 4
• Definition 5: Function Distance Matrix (FDM)
• Example 3
• Corollary 1: LBWY2023