We introduce generalized function-correcting partition codes (GFCPCs) that simultaneously protect multiple partitions of the message space against different numbers of errors. Given partitions with respective distance requirements, a GFCPC is a systematic encoding that guarantees, for each partition, a specified minimum Hamming distance between codewords whose messages lie in different blocks. This framework unifies and generalizes both function-correcting partition codes, which protect multiple functions with a common error-correction level, and function-correcting codes with data protection, which assign different levels of protection to data and a single function. We present a multi-step construction procedure for these codes and demonstrate it with some examples. We derive general upper and lower bounds on the optimal redundancy, including the upper bound which considers the join of different combinations of the partitions. We define the distance requirement matrix $\mathcal{D}$ for the GFCPCs and use it to characterize the optimal redundancy in terms of the shortest length of an associated $\mathcal{D}$-code. For two partitions of message space over the binary field, we establish improved lower bounds on the optimal redundancy under specific neighborhood conditions on the partitions. Through several examples, we demonstrate that the proposed framework can yield strictly smaller redundancy than both the sum of the individual FCPC redundancies and the redundancy of a single FCPC designed for the join partition with the highest distance (strongest protection required).

翻译：我们引入了广义函数修正分区编码（GFCPCs），该编码能同时保护消息空间的多个分区免受不同数量的错误影响。给定具有各自距离要求的多个分区，GFCPC是一种系统编码，能够保证对于每个分区，其消息位于不同区块的码字之间满足指定的最小汉明距离。该框架统一并推广了以下两类编码：在共同纠错级别下保护多个函数的函数修正分区编码，以及为数据与单一函数分配不同保护级别的带数据保护函数修正编码。我们提出了一种多步构造方法，并通过实例加以演示。我们推导了最优冗余的通用上界与下界，其中上界考虑了分区不同组合的联合。我们为GFCPC定义了距离需求矩阵$\mathcal{D}$，并利用该矩阵通过关联$\mathcal{D}$码的最短长度来刻画最优冗余。针对二元域上的两个消息空间分区，我们在特定分区邻域条件下建立了最优冗余的改进下界。通过多个实例，我们证明所提框架的冗余度严格小于以下两者：各独立FCPC冗余度之和，以及针对具有最高距离（所需最强保护）的联合分区设计的单一FCPC冗余度。