# Index Codes with Minimum Locality for Three Receiver Unicast Problems

## TL;DR

This paper establishes the optimal broadcast rate for three receiver unicast index coding problems with minimal locality, demonstrating the optimality of a known clique covering scheme through structural and information-theoretic analysis.

## Contribution

It provides tight converse results linking locality and broadcast rate, proving the optimality of a known index coding scheme at minimal locality for three receiver unicast problems.

## Key findings

- Optimal broadcast rate achieved at locality one.
- Structural properties of side information graphs derived.
- Known clique covering scheme proven optimal at minimal locality.

## Abstract

An index code for a broadcast channel with receiver side information is 'locally decodable' if every receiver can decode its demand using only a subset of the codeword symbols transmitted by the sender instead of observing the entire codeword. Local decodability in index coding improves the error performance when used in wireless broadcast channels, reduces the receiver complexity and improves privacy in index coding. The 'locality' of an index code is the ratio of the number of codeword symbols used by each receiver to the number message symbols demanded by the receiver. Prior work on locality in index coding have considered only single unicast and single-uniprior problems, and the optimal trade-off between broadcast rate and locality is known only for a few cases. In this paper we identify the optimal broadcast rate (including among non-linear codes) for all three receiver unicast problems when the locality is equal to the minimum possible value, i.e., equal to one. The index code that achieves this optimal rate is based on a clique covering technique and is well known. The main contribution of this paper is in providing tight converse results by relating locality to broadcast rate, and showing that this known index coding scheme is optimal when locality is equal to one. Towards this we derive several structural properties of the side information graphs of three receiver unicast problems, and combine them with information theoretic arguments to arrive at a converse.