On Function-Correcting Lee Metric Codes with Data Protection

2026-10-08

AI Summary

This study addresses the joint protection of data and function values under the Lee metric by proposing a framework of function-correcting codes with data protection. Methodologically, it integrates algebraic coding theory, Plotkin bound analysis, and combinatorial construction techniques to design an asymmetric error-correction mechanism that provides stronger protection for functions than for data. The main contributions include establishing theoretical lower and upper bounds on optimal redundancy, deriving explicit redundancy upper bounds for specific function classes, and demonstrating that the proposed framework naturally extends to the Hamming metric. Ultimately, this work achieves a unified coding scheme that simultaneously accommodates data and function error correction, thereby providing both a theoretical foundation and a constructive paradigm for asymmetric protection coding.

Abstract

Function-correcting codes are designed to protect the function values of a prescribed function against errors. Every error-correcting code that provides data protection inherently offers some degree of protection for functions defined on the data. In this work, we introduce a class of codes over $\mathbb{Z}_m$, termed function-correcting Lee metric codes with data protection (FCLMCs with data protection), which simultaneously provide error protection for both the data and the corresponding function values under the Lee metric. We consider codes that provide protection against a prescribed level of error for the function values that exceeds the level of protection guaranteed for the underlying data. We present a general construction of these codes and derive lower and upper bounds on the optimal redundancy, including a Plotkin-type lower bound. Furthermore, we derive explicit upper bounds on the redundancy of FCLMCs with data protection for several important classes of functions, including locally binary Lee functions, the Lee weight function, and the modular sum function. Finally, since the Lee metric coincides with the Hamming metric over $\mathbb{Z}_2$, all of our results remain valid over $\mathbb{Z}_2$ with respect to the Hamming metric.

Problem

• Function-correcting codes
• Lee metric
• Data protection
• Redundancy bounds
• Error correction

Innovation

• Function-correcting codes
• Lee metric
• Data protection
• Redundancy bounds
• Plotkin-type bound