On Function-Correcting Codes

TL;DR

This paper introduces a graph-based framework for analyzing function-correcting codes, deriving bounds on redundancy, and exploring their structure, especially for linear functions, thereby advancing understanding of systematic error correction.

Contribution

It provides a new graphical approach to bound redundancy in function-correcting codes and applies this to classical and linear codes, improving existing bounds and understanding.

Findings

Derived lower bounds on redundancy for function-correcting codes.

Showed the bounds are tight for certain parameters and classes of functions.

Simplified the Plotkin-like bound for linear functions.

Abstract

Function-correcting codes were introduced in the work "Function-Correcting Codes" (FCC) by Lenz et al. 2023, which provides a graphical representation for the problem of constructing function-correcting codes. We use this function dependent graph to get a lower bound on the redundancy required for function correction codes. By considering the function to be a bijection, such an approach leads to a lower bound on the redundancy required for classical systematic error correcting codes (ECCs). We propose a range of parameters for which the bound is tight. For single error correcting codes, we show that this bound is at least as good as a bound proposed by Zinoviev, Litsyn, and Laihonen in 1998. Thus, this framework helps to study systematic classical error correcting codes. Further, we study the structure of this function dependent graph for linear functions, which leads to bounds on the redundancy of linear-function correcting codes. We show that the Plotkin-like bound for function-correcting codes proposed by Lenz et.al 2023 is simplified for linear functions. We identify a class of linear functions for which an upper bound proposed by Lenz et al., is tight and also identify a class of functions for which coset-wise coding is equivalent to a lower dimensional classical error correction problem.