Codes protect data and related function values against errors in lee metric
On Function-Correcting Lee Metric Codes with Data Protection
Information Theory
Summary
Protecting information from errors can also protect the answers to questions asked about that information. The authors focus on special codes that not only fix errors in the raw data but also better protect the results of specific functions applied to that data. They create a new type of code that works over numbers with a certain error measure called the Lee metric, ensuring stronger protection for function results than the data itself. Their work includes ways to build these codes and mathematical limits on how efficient they can be. This also applies to the simpler case of binary data and the common Hamming error measure.
What this means in practice
- •For storage system engineers: Design storage that can correct data errors and ensure key computed values remain reliable using codes that work under the Lee metric.
- •For communication protocol developers: Create communication methods that protect both transmitted data and calculated results from noise using the proposed function-correcting Lee metric codes.
Authors
Gyanendra K. Verma, Abhay Kumar Singh
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.