Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. We first introduce a more general approach and a framework for function-correcting codes that incorporates data protection along with protection of function values. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph called minimum-distance graph to a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. To the best of our knowledge, this is the first instance of FCCs with a linear structure. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.

翻译：函数校正码旨在为基于数据计算的函数值提供错误保护。现有研究通常仅关注保护函数值本身，而忽略对底层数据的保护。本文提出一个通用框架，可同时为数据和函数值提供保护。由于保护数据本身即有助于保护函数值，我们重点研究函数值需要比数据本身更强保护的情形。首先，我们提出一种更通用的方法及框架，构建同时包含数据保护与函数值保护的函数校正码。提出了此类码的两步构造流程，并报告了具有数据保护的通用函数校正码最优冗余度的理论界。基于这些结果，我们展示了在不增加冗余度的前提下为现有函数校正码添加数据保护的实例。利用两步构造流程，我们为特定函数族（如局部有界函数和汉明权重函数）给出了具有数据保护的函数校正码的显式构造。通过将称为最小距离图的图结构与编码相关联，我们证明完美码和最大距离可分码无法为任何函数提供超越数据保护程度的额外函数值保护。随后聚焦于线性函数校正码，利用线性函数固有的结构特性，针对线性函数给出若干结论。据我们所知，这是首次提出具有线性结构的函数校正码。最后，我们将经典纠错编码理论中著名的普洛特金界和汉明界推广至具有数据保护的函数校正码。