带数据保护的函数纠错李度量码

中文摘要 AI 辅助

函数纠错码旨在保护指定函数的函数值免受错误影响。所有提供数据保护的纠错码，本质上都会对定义在数据上的函数提供一定程度的保护。本研究在$\boldsymbol{\text{Z}}_m$上引入一类名为带数据保护的函数纠错李度量码（FCLMCs with data protection）的码，这类码在李度量下可同时为数据及对应函数值提供纠错保护。我们考虑的码对函数值提供的错误保护级别，高于对底层数据的保护级别。我们给出这类码的通用构造方法，并推导最优冗余度的上下界，包括一个Plotkin型下界。此外，我们针对几类重要函数推导了带数据保护的FCLMCs的冗余度显式上界，包括局部二元李函数、李重量函数和模和函数。最后，由于李度量在$\boldsymbol{\text{Z}}_2$上与汉明度量一致，我们的所有结果在$\boldsymbol{\text{Z}}_2$上针对汉明度量同样有效。

英文摘要

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.

发表机构

• Indian Institute of Science（印度科学学院）
• Indian Institute of Technology (ISM) Dhanbad（印度技术学院（ISMT）丹巴德分校）

机构由 AI 辅助整理，请以论文原文为准。