An Upper Bound On the Size of Locally Recoverable Codes

TL;DR

This paper establishes a simple upper bound on the minimum distance of locally recoverable codes based on length, size, and locality, highlighting the optimality of binary Simplex codes and providing near-optimal constructions.

Contribution

It introduces a new, simple bound on the minimum distance of locally recoverable codes that depends on alphabet size, and identifies binary Simplex codes as optimal examples.

Findings

Binary Simplex codes meet the bound with equality.

New upper bound on minimum distance based on code parameters.

Constructed codes close to the theoretical bounds.

Abstract

In a {\em locally recoverable} or {\em repairable} code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage node as possible. In this paper, we bound the minimum distance of a code in terms of its length, size and locality. Unlike previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence the Simplex codes are the first example of a optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds.