We introduce function-correcting partition codes (FCPCs) that are a natural generalization of function-correcting codes (FCCs). A $t$-error function-correcting partition code is an $(\mathcal{P},t)$-encoding defined directly on a partition $\mathcal{P}$ of $\mathbb{F}_q^k$. For a partition $\mathcal{P}=\{P_1,P_2,\ldots,P_E\}$ a systematic mapping $\mathcal{C}_{\mathcal{P}} : \mathbb{F}_q^k \rightarrow \mathbb{F}_q^{k+r}$ is called a \emph{$(\mathcal{P},t)$-encoding} if for all $u\in P_i$ and $v\in P_j$ with $i\neq j$, $d\big(\mathcal{C}_{\mathcal{P}}(u), \mathcal{C}_{\mathcal{P}}(v)\big)\ge 2t+1.$ We show that any $t$-error correcting code for a function $f$, denoted by $(f,t)$-FCC 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 notion of partition redundancy gain and partition rate gain to measure the bandwidth saved by using a single FCPC for multiple functions instead of constructing separate FCCs for each function. We specialize this to linear functions via coset partition of the intersection of their kernels. Then, we associate a partition graph to 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. We showed the existence of a full-size clique in the partition graphs of weight partition and support partition. Finally, we introduce the notion of a block-preserving contraction for a partition, which helps reduce the problem of finding optimal redundancy for an FCPC. 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）的自然推广。一个$t$错误函数校正划分码是直接在$\mathbb{F}_q^k$的一个划分$\mathcal{P}$上定义的$(\mathcal{P},t)$编码。对于一个划分$\mathcal{P}=\{P_1,P_2,\ldots,P_E\}$，一个系统映射$\mathcal{C}_{\mathcal{P}} : \mathbb{F}_q^k \rightarrow \mathbb{F}_q^{k+r}$被称为\emph{$(\mathcal{P},t)$编码}，如果对于所有$u\in P_i$和$v\in P_j$且$i\neq j$，都有$d\big(\mathcal{C}_{\mathcal{P}}(u), \mathcal{C}_{\mathcal{P}}(v)\big)\ge 2t+1.$ 我们证明，任何函数$f$的$t$错误校正码（记为$(f,t)$-FCC）恰好就是关于由$f$诱导的定义域划分的FCPC，这使得这些码成为FCC的自然推广。我们利用定义域划分的并接来构造一个能同时保护多个函数的单一码。我们定义了划分冗余增益和划分速率增益的概念，用以衡量使用单一FCPC处理多个函数相比为每个函数单独构造FCC所节省的带宽。我们通过其核的交集的陪集划分，将这一概念专门应用于线性函数。接着，我们将一个划分图与$\mathbb{F}_q^k$的任何给定划分相关联，并证明在该图中存在一个合适的团，就能得到一组实现最优冗余的代表性信息向量。我们证明了在权重划分和支持划分的划分图中存在一个满规模的团。最后，我们引入了划分的块保持收缩的概念，这有助于简化寻找FCPC最优冗余的问题。我们观察到，FCPC自然地提供了一种形式的局部隐私，其意义在于只需向发送方揭示函数的定义域划分。