标题： 具有同质距离的功能校正码 显示英文标题

标题： Function-Correcting Codes with Homogeneous Distance

摘要： 函数校正码旨在减少在保护信息功能值免受错误时的冗余度。 作为在$ \mathbb{Z}_{4} $上汉明权和李权的推广，齐次权被用于有限环上的码。 在本文中，我们引入了具有齐次距离的函数校正码，记为 FCCHDs，它们扩展了具有汉明距离的函数校正码。 我们首先定义了$ D $-齐次距离码。 我们使用$ D $-齐次距离码来描述 FCCHDs 的最优冗余与某些矩阵$ D $的长度之间的关系。 通过这些关系，我们得到了一些函数下 FCCHDs 的最优冗余界。 此外，我们还构造了齐次权函数和齐次权分布函数的 FCCHDs。 特别地，我们在此论文中构造的一些码的冗余达到了最优冗余界。 显示英文摘要

摘要： Function-correcting codes are designed to reduce redundancy of codes when protecting function values of information against errors. As generalizations of Hamming weights and Lee weights over $ \mathbb{Z}_{4} $, homogeneous weights are used in codes over finite rings. In this paper, we introduce function-correcting codes with homogeneous distance denoted by FCCHDs, which extend function-correcting codes with Hamming distance. We first define $ D $-homogeneous distance codes. We use $ D $-homogenous distance codes to characterize connections between the optimal redundancy of FCCHDs and lengths of these codes for some matrices $ D $. By these connections, we obtain several bounds of the optimal redundancy of FCCHDs for some functions. In addition, we also construct FCCHDs for homogeneous weight functions and homogeneous weight distribution functions. Specially, redundancies of some codes we construct in this paper reach the optimal redundancy bounds.