Function-Correcting Partition Codes

Abstract

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.