Codes with locality for two erasures

Abstract

In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan et al, in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Turán graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from 2 erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.

On cooperative local repair in distributed storage

“…2) Comparison with the work in [24]: Recently, Prakash et al study codes which allow for local repair of 2 erasures [24]. In their model, they perform the repair of the two erasures in a successive manner, where a parity constraint of weight at mostr + 1 is used to repair each of the two erasures.…”

“…In their model, they perform the repair of the two erasures in a successive manner, where a parity constraint of weight at mostr + 1 is used to repair each of the two erasures. In [24], Prakash et al show that such codes have their rates upper bounded byr r+2 .…”

“…Since the upper boundr r+2 from [24] still applies to these codes, they exhibit almost optimal rate.…”

Binary Locally Repairable Codes from Complete Multipartite Graphs

“…부분접속수의 이론적 한계식 [2] 이 제시된 이후 부분접속 복구 부호에 대한 연구와 좀 더 일반화된 부분접속수에 대한 연구가 활발히 진행되고 있다 [3][4][5][6][7][8][9][10][11] . 특히 최근 임의의 노드 개가 손실된 경우 손실된 개의 노드들을 복구하기 위하여 이용되어야 하는 최소 노드의 개수를 -부분접속수(  )로 정의하 고, -부분접속수를 만족하는 부호 설계에 대한 연구 가 발표되었다 [11] .…”

Bounds on locally recoverable codes with multiple recovering sets

“…Recently codes that generalize ReedSolomon codes and achieve the bound (2) for any n were constructed in [12]. Other bounds on the distance of LRC codes appear in [1], [6]. A graph-theoretic proof of Theorem 1.1 was recently presented in [12].…”