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（FCC）旨在为数据上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，而忽略对底层数据的保护。本文提出一个通用框架，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，我们重点关注函数值需要比数据本身更强保护的情景。首先，我们引入一种更通用的方法及框架，将数据保护与函数值保护相结合构建函数校正码。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FCC最优冗余度的理论界。基于这些结果，我们通过示例证明，可在不增加冗余度的前提下为现有FCC添加数据保护功能。利用两步构造法，我们针对特定函数族（如局部有界函数和汉明权重函数）给出了具有数据保护的FCC显式构造。通过将最小距离图与编码关联，证明了完美码和最大距离可分（MDS）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FCC，利用线性函数的内在结构特性，给出了针对线性函数的相关结果。据我们所知，这是首次提出具有线性结构的FCC。最后，我们将经典纠错编码理论中广为人知的普洛特金界和汉明界推广至具有数据保护的FCC。