Function-Correcting Partition codes

2026-01-10

arXiv.org

AI Summary

This work addresses the challenge of efficiently providing fault-tolerant protection for the results of multiple functions in communication systems by proposing Function-Correcting Partition Codes (FCPC). The approach extends traditional function-correcting codes into a partitioned coding framework over the information space, leveraging domain partitioning, systematic encoding mappings, and partition graph modeling. By exploiting the coset structure induced by intersections of kernels of linear functions and applying clique analysis from graph theory, FCPC enables joint protection of multiple functions. The design incorporates block-preserving compression to optimize redundancy, guided by newly introduced metrics—partition redundancy gain and rate gain. Theoretically, FCPC is shown to be equivalent to conventional function-correcting codes while achieving optimal redundancy under weight- and support-based partitions, thereby significantly reducing overhead.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• function-correcting codes
• partition codes
• error correction
• multiple functions
• redundancy reduction

Innovation

• function-correcting codes
• partition codes
• error correction
• clique in partition graph
• redundancy gain