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

TL;DR

This paper introduces a new class of pliable index coding problems called group complete-$S$ PICOD, providing optimal coding schemes and bounds for certain parameters, advancing the understanding of broadcast rate limits.

Contribution

It defines the group complete-$S$ PICOD problem, develops a multi-stage coding scheme, and establishes optimality bounds for specific parameter ranges.

Findings

Achieves optimal broadcast rates for singleton $S$ and certain $t$, $g$, $s$ values.

Provides lower bounds on broadcast rate using acyclic subgraph bounds.

Shows the scheme's optimality within restricted code classes for other parameters.

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.