We introduce function-correcting partition codes (FCPCs), which are a natural generalization of function-correcting codes (FCCs). An FCPC is defined directly on a partition of the message space, rather than on a specific target function. We show that any FCC for a function $f$ is exactly an FCPC with respect to the domain partition induced by $f$, which makes these codes a natural generalization of FCCs. We use the join of domain partitions to construct a single code that protects multiple functions simultaneously. We define the notions of partition gains to measure the bandwidth saved by using a single FCPC for multiple functions instead of constructing separate FCCs for each function. We derive general lower and upper bounds on the redundancy of such FCPCs and illustrate the achievable gains through examples. We specialize this concept of using single code for protecting multiple functions to linear functions via coset partition of the intersection of their kernels. We also present explicit FCPC constructions for locally bounded partitions and grouped weight partitions. Then, we associate a partition graph with any given partition of $\mathbb{F}_q^k$, and show that the existence of a suitable clique in this graph yields a set of representative information vectors that achieves the optimal redundancy. Using the existence of a full-size clique in the weight partition and support partition, we obtain lower and upper bounds on the optimal redundancy of FCPCs for these partitions. We introduce the notion of a block-preserving contraction for a partition, which helps reduce the problem size of finding optimal redundancy for an FCPC. We further show that such a contraction exists for all weight-based partitions. Finally, we observe that FCPCs naturally provide a form of partial privacy in the sense that only the domain partition of the function needs to be revealed to the transmitter.

翻译：本文引入了函数校正划分码（FCPC），它是函数校正码（FCC）的一种自然推广。FCPC直接定义在消息空间的划分上，而非针对特定目标函数。我们证明，对于任意函数 $f$ 的FCC，恰好就是关于由 $f$ 诱导的域划分的FCPC，这使得这些码成为FCC的自然推广。我们利用域划分的连接来构造一个能同时保护多个函数的单一码。我们定义了划分增益的概念，用以衡量使用单一FCPC保护多个函数，而非为每个函数分别构造FCC所节省的带宽。我们推导了此类FCPC冗余度的一般上下界，并通过示例说明了可实现的增益。我们将这种使用单一码保护多个函数的概念专门应用于线性函数，通过其核的交集的陪集划分来实现。我们还针对局部有界划分和分组权重划分提出了显式的FCPC构造。接着，我们将一个划分图与 $\mathbb{F}_q^k$ 的任意给定划分相关联，并证明该图中存在一个合适的团，即可得到一组能达到最优冗余度的代表性信息向量。利用权重划分与支撑划分中存在满尺寸团的性质，我们得到了针对这些划分的FCPC最优冗余度的上下界。我们引入了划分的块保持收缩概念，这有助于减小寻找FCPC最优冗余度的问题规模。我们进一步证明，所有基于权重的划分都存在这样的收缩。最后，我们观察到，FCPC自然地提供了一种形式的局部隐私，即仅需向发送方揭示函数的域划分。