# Locality in Index Coding for Large Min-Rank

## TL;DR

This paper investigates the optimal locality of linear index codes in networks with min-rank just below the number of receivers, providing new trade-offs and structural insights for efficient decoding.

## Contribution

It derives the optimal rate-locality trade-off for specific index coding problems, introducing new structural results for locally decodable codes.

## Key findings

- Optimal trade-off between rate and locality for directed cycle graphs.
- Scalar linear coding achieves optimal trade-off for certain problem classes.
- New structural results on locally decodable index codes.

## Abstract

An index code is said to be locally decodable if each receiver can decode its demand using its side information and by querying only a subset of the transmitted codeword symbols instead of observing the entire codeword. Local decodability can be a beneficial feature in some communication scenarios, such as when the receivers can afford to listen to only a part of the transmissions because of limited availability of power. The locality of an index code is the ratio of the maximum number of codeword symbols queried by a receiver to the message length. In this paper we analyze the optimum locality of linear codes for the family of index coding problems whose min-rank is one less than the number of receivers in the network. We first derive the optimal trade-off between the index coding rate and locality with vector linear coding when the side information graph is a directed cycle. We then provide the optimal trade-off achieved by scalar linear coding for a larger family of problems, viz., problems where the min-rank is only one less than the number of receivers. While the arguments used for achievability are based on known coding techniques, the converse arguments rely on new results on the structure of locally decodable index codes.