Function-Correcting Codes with Homogeneous Distance

TL;DR

This paper introduces function-correcting codes with homogeneous distance (FCCHDs) over finite rings, extending traditional codes with Hamming distance, and establishes bounds and constructions for optimal redundancy.

Contribution

It defines D-homogeneous distance codes and explores their connection to optimal redundancy in function-correcting codes, including new bounds and constructions.

Findings

Derived bounds for the optimal redundancy of FCCHDs.

Constructed FCCHDs that reach optimal redundancy bounds.

Established connections between code parameters and matrix functions.

Abstract

Function-correcting codes are designed to reduce redundancy of codes when protecting function values of information against errors. As generalizations of Hamming weights and Lee weights over $ \mathbb{Z}_{4} $, homogeneous weights are used in codes over finite rings. In this paper, we introduce function-correcting codes with homogeneous distance denoted by FCCHDs, which extend function-correcting codes with Hamming distance. We first define $ D $-homogeneous distance codes. We use $ D $-homogenous distance codes to characterize connections between the optimal redundancy of FCCHDs and lengths of these codes for some matrices $ D $. By these connections, we obtain several bounds of the optimal redundancy of FCCHDs for some functions. In addition, we also construct FCCHDs for homogeneous weight functions and homogeneous weight distribution functions. Specially, redundancies of some codes we construct in this paper reach the optimal redundancy bounds.