Optimal Linear Codes with a Local-Error-Correction Property

TL;DR

This paper extends the concept of local-symbol recovery in linear codes to include error correction in local parities, providing tight bounds on minimum distance and constructing optimal codes for distributed storage applications.

Contribution

It introduces a new class of optimal linear codes with local-error-correction capabilities, expanding previous notions of locality to handle local errors.

Findings

Derived tight bounds on minimum distance for local-error-correction codes.

Constructed codes that achieve optimality with respect to local error correction.

Provided an upper bound on the minimum distance of concatenated codes.

Abstract

Motivated by applications to distributed storage, Gopalan \textit{et al} recently introduced the interesting notion of information-symbol locality in a linear code. By this it is meant that each message symbol appears in a parity-check equation associated with small Hamming weight, thereby enabling recovery of the message symbol by examining a small number of other code symbols. This notion is expanded to the case when all code symbols, not just the message symbols, are covered by such "local" parity. In this paper, we extend the results of Gopalan et. al. so as to permit recovery of an erased code symbol even in the presence of errors in local parity symbols. We present tight bounds on the minimum distance of such codes and exhibit codes that are optimal with respect to the local error-correction property. As a corollary, we obtain an upper bound on the minimum distance of a concatenated code.