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).