On taking advantage of multiple requests in error correcting codes

TL;DR

This paper explores the trade-offs in error correcting codes when recovering multiple message symbols with limited queries, introducing robust batch codes and showing repetition remains optimal for large r.

Contribution

It introduces robust batch codes for recovering multiple symbols with limited queries and demonstrates that repetition is optimal even for large r values.

Findings

Repetition remains optimal for large r in robust batch codes.

Introduces the concept of robust batch codes as a generalization of MDS properties.

Analyzes the effectiveness of different local decoding strategies.

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 $r = \Omega(k)$.