The Single-Uniprior Index-Coding Problem: The Single-Sender Case and The Multi-Sender Extension

TL;DR

This paper investigates a specialized index coding problem with single-message receivers and multiple senders, proposing algorithms to bound and achieve optimal code lengths, especially in cases with no message overlap among senders.

Contribution

It introduces pruning and appending techniques for bounding the optimal index codelength in single- and multi-sender scenarios, with exact solutions when senders have disjoint message sets.

Findings

Pruning algorithm provides a lower bound on codelength for single sender.

Appending technique and cyclic codes give bounds for multiple senders.

Bounds match when senders have no common messages, achieving optimal codelength.

Abstract

Index coding studies multiterminal source-coding problems where a set of receivers are required to decode multiple (possibly different) messages from a common broadcast, and they each know some messages a priori. In this paper, at the receiver end, we consider a special setting where each receiver knows only one message a priori, and each message is known to only one receiver. At the broadcasting end, we consider a generalized setting where there could be multiple senders, and each sender knows a subset of the messages. The senders collaborate to transmit an index code. This work looks at minimizing the number of total coded bits the senders are required to transmit. When there is only one sender, we propose a pruning algorithm to find a lower bound on the optimal (i.e., the shortest) index codelength, and show that it is achievable by linear index codes. When there are two or more senders, we propose an appending technique to be used in conjunction with the pruning technique to give a lower bound on the optimal index codelength; we also derive an upper bound based on cyclic codes. While the two bounds do not match in general, for the special case where no two distinct senders know any message in common, the bounds match, giving the optimal index codelength. The results are expressed in terms of strongly connected components in directed graphs that represent the index-coding problems.