Locally Decodable Index Codes

TL;DR

This paper studies the trade-off between broadcast rate and local decodability in index coding, providing optimal solutions for specific graph structures and general techniques for designing locally decodable codes.

Contribution

It characterizes the optimal rate-locality trade-off for certain graph classes and introduces methods to design locally decodable index codes for general problems.

Findings

Optimal trade-off for directed cycle graphs with vector linear codes.

Optimal trade-off for graphs with minrank one less than the number of nodes.

Techniques for designing locally decodable codes for arbitrary graphs.

Abstract

An index code for broadcast channel with receiver side information is locally decodable if each receiver can decode its demand by observing only a subset of the transmitted codeword symbols instead of the entire codeword. Local decodability in index coding is known to reduce receiver complexity, improve user privacy and decrease decoding error probability in wireless fading channels. Conventional index coding solutions assume that the receivers observe the entire codeword, and as a result, for these codes the number of codeword symbols queried by a user per decoded message symbol, which we refer to as locality, could be large. In this paper, we pose the index coding problem as that of minimizing the broadcast rate for a given value of locality (or vice versa) and designing codes that achieve the optimal trade-off between locality and rate. We identify the optimal broadcast rate corresponding to the minimum possible value of locality for all single unicast problems. We present new structural properties of index codes which allow us to characterize the optimal trade-off achieved by: vector linear codes when the side information graph is a directed cycle; and scalar linear codes when the minrank of the side information graph is one less than the order of the problem. We also identify the optimal trade-off among all codes, including non-linear codes, when the side information graph is a directed 3-cycle. Finally, we present techniques to design locally decodable index codes for arbitrary single unicast problems and arbitrary values of locality.