A Generic Transformation to Generate MDS Storage Codes with $δ$-Optimal Access Property

Abstract

For high-rate maximum distance separable (MDS) codes, most of them are designed to optimally repair a single failed node by connecting all the surviving nodes. However, in practical systems, sometimes not all the surviving nodes are available. To facilitate the practical storage system, a few constructions of (n, k) MDS codes with the property that any single failed node can be optimally repaired by accessing any d surviving nodes have been proposed, where d ∈ [k + 1 : n − 1). However, all of them either have large sub-packetization levels or are not explicit for all the parameters. To address these issues, we propose a generic transformation that can convert any (n ′ , k ′ ) MDS code to another (n, k) MDS code with the optimal repair property and optimal access property for an arbitrary set of two nodes, while the repair efficiency of the remaining n − 2 nodes can be kept. By recursively applying the generic transformation to a scalar MDS code multiple times, we get an MDS code with the optimal repair property and the optimal access property for all nodes, which outperforms previous known MDS codes in terms of either the sub-packetization level or the flexibility of the parameters.

Optimal Repair/Access MDS Array Codes with Multiple Repair Degrees

“…An (n, k) MDS code with the θ-optimal repair property is preferred, i.e., any failed node can be repaired by downloading N θ symbols from each of the d = k + θ − 1 helper nodes. In this paper, similarly to that in [11], when repairing a failed node i ∈ [0 : n), the N θ symbols downloaded from each helper node j ∈ H is represented by R i,θ f j , where H denotes the indices set of the d helper nodes and the N θ × N matrix R i,θ of full rank is called the θ-repair matrix of node i. In addition, the code is preferred to have the θ-optimal access property, i.e., when repairing a failed node i ∈ [0 : n), the amount of accessed data attains the lower bound in (1).…”

“…t,i is well defined for (11). The following two examples illustrate the whole process of our method.…”

“…In other words, when d z surviving nodes are connected, we use the equations obtained by multiplying S i,δs on both sides of the equations in (11) to recover the data stored at node i, i.e., (5) into the above LSEs, we then get…”

“…Particularly, in this paper, the MDS array code with the optimal access property and optimal sub-packetization level proposed in [20] is called VBK code. More recently, in [11], Liu et al proposed an explicit construction of high-rate (n, k) MDS array code with the θ-optimal access property for all nodes, where the sub-packetization level is θ ⌈ n 2 ⌉ , which is between that of the YB codes 1, 2 [23] and the ones proposed in [17], [20].…”