On Taking Advantage of Multiple Requests in Error Correcting Codes

Abstract

In most notions of locality in error correcting codes-notably locally recoverable codes (LRCs) and locally decodable codes (LDCs)-a decoder seeks to learn a single symbol of a message while looking at only a few symbols of the corresponding codeword. However, suppose that one wants to recover r > 1 symbols of the message. The two extremes are repeating the single-query algorithm r times (this is the intuition behind LRCs with availability, primitive multiset batch codes, and PIR codes) or simply running a global decoding algorithm to recover the whole thing. In this paper, we investigate what can happen in between these two extremes: at what value of r does repetition stop being a good idea?In order to begin to study this question we introduce robust batch codes, which seek to find r symbols of the message using m queries to the codeword, in the presence of erasures. We focus on the case where r = m, which can be seen as a generalization of the MDS property. Surprisingly, we show that for this notion of locality, repetition is optimal even up to very large values of r = Ω(k).

A locality-based approach for coded computation

“…Existing literature on locality of codes [25], [26], [27], [28], [29], [30] usually considers a code where each code symbol appears exactly once. The associated codeword reflects this convention.…”

“…To do so, we define a new notion of locality for computation, denoted computational locality, via locality properties of an appropriately defined code. The topic of the locality of codes has a rich literature [25], [26], [27], [28], [29], [30]. For instance, there are well known "local recovery schemes"-procedures to recover one erased code symbol by querying a few other code symbols, for several classes of codes such as Reed-Muller codes.…”

Batch Codes from Affine Cartesian Codes and Quotient Spaces

“…This is similar to Private information retrieval (PIR) codes, which differ in that t duplicates of the same element must be reconstructed. Other schemes dealing with multiple requests are addressed in [11]. For batch and PIR codes where the queries do not all necessarily occur at the same time, see [12].…”