# Decentralized Pliable Index Coding

## TL;DR

This paper studies a decentralized variant of pliable index coding, characterizing its capacity for specific cases and showing that optimal code lengths match the centralized case, but with different coding strategies.

## Contribution

It introduces the decentralized PICOD problem, derives capacity results for certain classes, and highlights differences in coding strategies from the centralized version.

## Key findings

- Capacity matches centralized PICOD for certain classes.
- Optimal code length is the same as centralized PICOD in these cases.
- Decentralized PICOD uses sparse MDS and vector linear codes instead of scalar codes.

## Abstract

This paper introduces the ${\it decentralized}$ Pliable Index CODing (PICOD) problem: a variant of the Index Coding (IC) problem, where a central transmitter serves ${\it pliable}$ users with message side information; here, pliable refers to the fact that a user is satisfied by decoding ${\it any}$ $t$ messages that are not in its side information set. In the decentralized PICOD, a central transmitter with knowledge of all messages is not present, and instead users share among themselves massages that can only depend on their local side information set. This paper characterizes the capacity of two classes of decentralized complete--$S$ PICOD$(t)$ problems with $m$ messages (where the set $S\subset[m]$ contains the sizes of the side information sets, and the number of users is $n=\sum_{s\in S}\binom{m}{s}$, with no two users having the same side information set): (i) the consecutive case: $S=[s_\min:s_\max]$ for some $0 \leq s_\min\leq s_\max \leq m-t$, and (ii) the complement-consecutive case: $S=[0:m-t]\backslash[s_\min:s_\max]$, for some $0 < s_\min\leq s_\max < m-t$. Interestingly, the optimal code-length for the decentralized PICOD in those cases is the same as for the classical (centralized) PICOD counterpart, except when the problem is no longer pliable, that is, it reduces to an IC problem where every user needs to decode all messages not in its side information set. Although the optimal code-length may be the same in both centralized and decentralized settings, the actual optimal codes are not. For the decentralized PICOD, sparse Maximum Distance Separable (MDS) codes and vector linear index codes are used (as opposed to scalar linear codes).