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.

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

“…In this paper, we focus only on all-symbol locality and therefore drop the specification. A repair set R of an LRC is a set of coordinates such that any δ − 1 code symbols c i with i ∈ R can be obtained from the remaining code symbols with indices in R. Other extensions of the locality property include codes with availability [8], sequential repair of several erasures [9], cooperative repair [10], local repair on graphs [11] and many others.…”

A Tight Rate Bound and Matching Construction for Locally Recoverable Codes With Sequential Recovery From Any Number of Multiple Erasures

“…It can be seen that the upper bound on rate given in Theorem 2 is of the form given in Conjecture 1. We prove the conjecture in full here i.e., we will prove in Section VIII that the upper bound in Theorem 2 is also achievable by constructing binary codes that achieve the upper bound on code rate for any r ≥ 3 and any t. The upper bound on rate given in Theorem 2, for t = 2, 3, coincides with the upper bound given in [12] and [30] respectively. For t ≥ 4, the upper bound on rate given in Theorem 2 is new.…”

“…In the last section, we saw that the p-c matrix of a rate-optimal seq-LRC can be assumed without loss of generality, to have the staircase form appearing in equations (11) and (12). It will be shown in the present section, just as was done in the case of the examples presented in Section III, that this form of p-c matrix leads to a graphical representation of the code.…”

“…When (r + 1) n, the bound (1) is not achievable in general. The bound in (1) has been improved upon and tighter upper bounds can be found in [12], [13], [14], [15]. Constructions of LRC with field size exponential in n achieving the tighter upper bound on minimum distance derived in [13] for the case of n 1 > n 2 where n 1 = n r+1 , n 2 = n 1 (r + 1) − n can also be found in [13].…”

“…The sequential approach to recovery from multiple erasures was introduced by Prakash et al [12] and has been further investigated in [31], [32], [33], [34], [35], [30], [1], [2] as discussed below. a) Two Erasures: Seq-LRC with t = 2 are considered in [12] (see also [32]) where a tight upper bound on the rate and a matching construction achieving the upper bound on rate for t = 2 is provided. A lower bound on block length and a construction achieving the lower bound on block length for t = 2 is provided in [32].…”