Group Complete-$\{s\}$ Pliable Index Coding

TL;DR

This work extends pliable index coding to a group-structured setting by introducing g-group complete-$S$ PICOD($t$) and solving the singleton $S$ case via a multi-stage, MDS-based achievability scheme. A graph-theoretic MAIS bound is developed and employed to derive converse results, yielding exact rate characterizations in regimes determined by $t$, $m$, $g$, and $s$, and showing optimality of the proposed scheme for $t > (g-1)(m-s)$. The results generalize prior complete-$S$ PICOD analyses (including the $g=1$ case studied by Liu and Tuninetti) and provide a concrete, nonconvex achievable rate framework due to the absence of time-sharing. Overall, the paper advances understanding of group-structured pliable index coding and furnishes precise rate results and coding strategies for practical broadcast scenarios with grouped side information.

Abstract

This paper introduces a novel class of PICOD($t$) problems referred to as $g$-group complete-$S$ PICOD($t$) problems. It constructs a multi-stage achievability scheme to generate pliable index codes for group complete PICOD problems when $S = \{s\}$ is a singleton set. Using the maximum acyclic induced subgraph bound, lower bounds on the broadcast rate are derived for singleton $S$, which establishes the optimality of the achievability scheme for a range of values for $t$ and for any $g$ and $s$. For all other values, it is shown that the achievability scheme is optimal among the restricted class of broadcast codes.

Figures (1)

• Figure 1: The achievable rate given in Theorem \ref{['thm:achievable']}

Theorems & Definitions (16)

• Lemma 1
• Corollary 1
• Corollary 2
• Lemma 2
• Remark 1
• Corollary 3
• Theorem 1
• Lemma 3
• Lemma 4
• Lemma 5