On Locally Decodable Index Codes

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.

Embedded Index Coding

“…This can be seen as a generalization of Instantly Decodable Network Codes [14] which have been studied with similar motivation (see Remark 1). Task based solutions are also related to Locally Decodable Index Codes [24] (see Remark 2).…”

“…An index coding solution has locality r if each node uses at most r received symbols to decode any message symbol. There is tradeoff between optimal broadcast rate and locality of solutions for a given index coding problem [24]. When r = 1, locally decodable index codes are a special case of task-based schemes, although the notions diverge for more general r (see Remark 2).…”

Index Codes with Minimum Locality for Three Receiver Unicast Problems

“…An index code is locally decodable if every user can decode its demand by using its side information and by observing only a subset of the transmitted codeword symbols (instead of observing the entire codeword) [10]. The locality of an index code is the ratio of the maximum number of codeword symbols observed by any receiver to the number of message symbols demanded by the receiver [11]. The objective of designing…”

“…Prior work on locally decodable index codes have considered single unicast index coding problems [10]- [12], problems where each receiver demands exactly one message [14], and a family of index coding problems called single uniprior 1 [13], [15]. The exact characterization of the optimal trade-off between rate and locality is known in only a few cases, all of them being single unicast problems: optimal rate among nonlinear codes for single unicast problems and locality equal to one [11], rate-locality trade-off among linear codes for singleunicast problems when min-rank is one less than the number of receivers [16].…”

Locality in Index Coding for Large Min-Rank

“…In such scenarios, it is desirable to use locally decodable index codes [8], which provide reductions in broadcast rate while requiring the receivers to query only a part of the transmitted codeword. The locality of an index code Dr is the ratio of the number of codeword symbols queried by a receiver to the number of message symbols it demands [9]. The objective of locally decodable index coding is to minimize both broadcast rate and locality simultaneously, and achieve the optimal trade-off between these two parameters.…”