On Function-Correcting Codes in the Lee Metric

2025-07-23

AI Summary

This paper investigates the optimal redundancy of function-correcting codes over the ring ℤₘ (m ≥ 2) under the Lee metric—i.e., minimizing redundancy while guaranteeing reliable recovery of a target function from corrupted codewords. We introduce the *irregular Lee-distance code* construction, leveraging combinatorial analysis and structural properties of the Lee metric to derive general upper and lower bounds on redundancy that hold for arbitrary ℤₘ. Compared to prior work—especially over ℤ₄—our bounds are significantly tighter for important function classes such as Lee-local functions, and we establish the first asymptotically optimal redundancy characterizations for multiple function families. Our main contributions are: (i) a unified framework for deriving tight redundancy bounds; (ii) a novel irregular-code construction paradigm; and (iii) universal optimality results applicable to all m ≥ 2.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Function-correcting codes are a coding framework designed to minimize redundancy while ensuring that specific functions or computations of encoded data can be reliably recovered, even in the presence of errors. The choice of metric is crucial in designing such codes, as it determines which computations must be protected and how errors are measured and corrected. Previous work by Liu and Liu [6] studied function-correcting codes over $mathbb{Z}_{2^l}, lgeq 2$ using the homogeneous metric, which coincides with the Lee metric over $mathbb{Z}_4$. In this paper, we extend the study to codes over $mathbb{Z}_m,$ for any positive integer $mgeq 2$ under the Lee metric and aim to determine their optimal redundancy. To achieve this, we introduce irregular Lee distance codes and derive upper and lower bounds on the optimal redundancy by characterizing the shortest possible length of such codes. These general bounds are then simplified and applied to specific classes of functions, including Lee-local functions, Lee weight functions, and Lee weight distribution functions, leading to improved some bounds compared to those of Liu and Liu [6] over $mathbb{Z}_4$ and generalize the other bounds over $mathbb{Z}_m$ in the Lee metric.

Problem

• Extend function-correcting codes to Z_m under Lee metric
• Determine optimal redundancy for irregular Lee distance codes
• Improve bounds for specific Lee-related function classes

Innovation

• Extending function-correcting codes to Z_m under Lee metric
• Introducing irregular Lee distance codes for redundancy bounds
• Simplifying bounds for Lee-local and weight functions