Function-Correcting Codes for Linear and Locally Bounded Functions Over a Finite Chain Ring

TL;DR

This paper extends the theory of function-correcting codes over chain rings, introducing new bounds and constructions for locally bounded and linear functions, with implications for code efficiency and optimality.

Contribution

It introduces new bounds and explicit constructions for function-correcting codes over chain rings, focusing on locally bounded and linear functions, and analyzes their optimality.

Findings

Derived a Plotkin-like bound for irregular homogeneous distance codes over 

Established bounds on the redundancy of function-correcting codes using locality properties

Provided explicit constructions demonstrating the bounds and discussing their tightness

Abstract

In this paper, we further extend the study of function-correcting codes in the homogeneous metric over a chain ring $\mathbb{Z}_{2^s}$ for broader classes of functions, namely, locally bounded functions and linear functions, and for weight functions, modular sum functions. e define locally bounded functions in the homogeneous metric over $\mathbb{Z}_{2^s}^k$ and investigate the locality of weight functions. We derive a Plotkin-like bound for irregular homogeneous distance code over $\mathbb{Z}_4$, which improves the existing bound. Using locality properties of functions, we establish upper and lower bounds on the optimal redundancy. We provide several explicit constructions of function-correcting codes for locally bounded functions, weight functions, and weight distribution functions. Using these constructions, we further discuss the tightness of the derived bound. We explicitly derive a Plotkin-like bound for linear function-correcting codes that reduces to the classical Plotkin bound when the linear function is bijective, we further discuss a construction of function-correcting linear codes over $\mathbb{Z}_{2^s}$.