Function-Correcting Partition codes

TL;DR

The notion of partition redundancy gain and partition rate gain is defined to measure the bandwidth saved by using a single FCPC for multiple functions instead of constructing separate FCCs for each function.

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.

Figures (8)

• Figure 1: Illustration of the function-correcting partition code (FCPC) setup. A message $u$ is transmitted with redundancy $p$ over a noisy channel. The transmitter has access only to a partition of the message domain and does not know the complete function desired by the receiver. Using the received word $y$ and the knowledge of the function $f$, the receiver correctly recovers the function value $f(u)$.
• Figure 2: Multiple receivers and single–receiver multi-function settings.
• Figure 3: Partition graph for $\mathcal{P}_1$.
• Figure 4: Partition graph for $\mathcal{P}_2$.
• Figure 5: A clique of graph $G_{\mathcal{P}_2}$.

Theorems & Definitions (86)

• Definition 1: Partition
• Definition 2: Refinement of a partition
• Definition 3: Coarsest common refinement or join
• Example 1
• Definition 4: Graph
• Definition 5: Clique
• Definition 6: $t$-partite Graph
• Definition 7: Function-correcting codes
• Definition 8: Distance Requirement Matrix (DRM)
• Definition 9: Irregular-distance code or $\mathcal{D}$-code