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

2026-03-15

AI Summary

This study addresses the design of functional error-correcting codes for locally bounded functions over the finite chain ring ℤ_{2^s} under the homogeneous metric, with a focus on linear functions. By analyzing the locality of such functions, the work derives upper and lower bounds on redundancy and proposes a unified framework for constructing error-correcting codes tailored to this setting. The main contributions include the first extension of the classical Plotkin bound to non-bijective linear functions, an improved Plotkin bound for irregular homogeneous-metric codes over ℤ₄, and several explicit constructions of functional error-correcting codes that demonstrate the tightness of the derived bounds.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $\mathbb{Z}_{2^s}$.

Problem

• function-correcting codes
• locally bounded functions
• linear functions
• homogeneous metric
• finite chain ring

Innovation

• function-correcting codes
• homogeneous metric
• locally bounded functions
• Plotkin-type bound
• linear functions