Optimal Linear Coding Schemes for the Secure Decentralized Pliable Index Coding Problem

TL;DR

This paper investigates secure decentralized pliable index coding with circular side information, identifying infeasible cases for linear codes and proposing schemes that approach the optimal code length within a constant gap.

Contribution

It characterizes infeasible cases for linear coding and develops near-optimal linear schemes for the remaining instances in secure decentralized pliable index coding.

Findings

Identified cases where linear codes cannot achieve security and decodability.

Proposed linear coding schemes that are within a constant gap of the converse bound.

Established the limitations and potential of linear codes in secure decentralized settings.

Abstract

We study the secure decentralized Pliable Index CODing (PICOD) problem with circular side information sets at the users. The security constraint forbids every user to decode more than one message while a decentralized setting means there is no central transmitter in the system. Compared to the secure but centralized version of the problem, a converse bound from one of our previous works showed a factor of three difference in optimal code length under the constraint of linear encoding. In this paper, we first list the linearly infeasible cases, that is, problems where no linear code can simultaneously achieve both correctness/decodability and security. Then, we propose linear coding schemes for all remaining cases and show that their attained code length is to within an additive constant gap from our converse bound.