A generic transformation for optimal repair bandwidth and rebuilding access in MDS codes

An Explicit Construction of Systematic MDS Codes with Small Sub-Packetization for All-Node Repair

“…1 is presented in [4], [5]. Note that in the original presentation the indexing of the arrays is from 1 to r but in order to synchronize with the transformation of Li et al [9] here we use the indexing of the arrays from 0 to r − 1. The set of all symbols in d j is partitioned in disjunctive subsets where at least one subset has ⌈ α /r⌉ number of elements.…”

“…, Pr−1 that have not been read in Step 2; 7: Access and transfer (r − 1)⌈ α /r 2 ⌉ paired symbols pi,j from the v-th instance; 8: Repair ai,j by solving paired r × r linear systems of equations. 9: end for…”

“…They are access-optimal (access and transfer the same amount of data). We combine the framework proposed by Li et al [9] and the family of MDS codes called HashTag codes [5]. Compared to the work by Li et al [9] where they focus on MSR codes with the maximal sub-packetization level α = r ⌈ n r ⌉ , we construct explicit codes for the whole range of sub-packetization levels 4 ≤ α ≤ r ⌈ n r ⌉ motivated by the practical importance of codes with small sub-packetization levels.…”

Erasure coding for distributed storage: an overview

“…Essentially the same construction was independently rediscovered in [39] from a different coupled-layer perspective, where layers of an arbitrary MDS codes are coupled by a simple pairwise coupling transform to yield an MSR code. Just prior to the appearance of these two papers, in an earlier version of [40], the authors show how a systematic MSR code can be converted into an MSR code by increasing the sub-packetization level by a factor of r = (n − k) using a pairwise symbol transformation. This result is then extended in [40], to present a technique that takes an MDS code, increases sub-packetization level by a factor of r and converts it into a code in which the optimal repair of r nodes can be carried out.…”

Systematic Construction of MDS Codes with Small Sub-packetization Level and Near Optimal Repair Bandwidth

“…In this paper, we aim to construct high-rate MDS codes that have both small sub-packetization level and near optimal repair bandwidth for general parameters n and k, while over a small finite field F q . We notice that there exist abundant high-rate MDS codes with the optimal repair bandwidth but require a large sub-packetization level in the literature [3]- [9], [11]- [13], [19], which intrigue us to think whether can we construct high-rate MDS codes that have both small sub-packetization level and near optimal repair bandwidth by using the abundant high-rate MDS codes with the optimal repair bandwidth. Partly motivated by the work in [3], we present a powerful transformation that can convert any MDS code into another MDS code with much longer code length, such that the new MDS code slightly increase the repair bandwidth but can keep the sub-packetization level as that of the original MDS code, or equivalently the generic transformation can reduce the sub-packetization level N of the original codes with respect to the same code length n. By directly applying the generic transformation to several known high-rate MDS codes with the optimal repair bandwidth, we get four high-rate (n, k) MDS codes that have both small sub-packetization level N and near optimal repair bandwidth, with two of them are explicit and the required field sizes are comparable to the code length n. Besides, we propose another new MDS code that have small sub-packetization level, near optimal repair bandwidth, and the optimal update property, while the required field size is also comparable to the code length n. The obtained MDS codes outperform the first MDS code construction in [18] in terms of the field size and outperform the first code in both [12] and [19], the second MDS code construction in [18] in terms of the sub-packetization level.…”