Two Generic Constructions of MDS Array Codes With Optimal Repair Bandwidth From Two Special Sets

Abstract

The maximum distance separable (MDS) codes are the optimal codes to achieve Singleton bound, providing maximum error tolerance under a given number of parity nodes. Ye and Barg leveraged permutation matrices and Reed-Solomon type codes to devise 7 explicit constructions for constructing MDS array codes with optimal repair property (as known as MSR codes) or even optimal access property. Drawing inspiration from these explicit constructions, we provide two generic constructions for constructing MSR codes from high-rate MDS codes or MDS array codes. In this paper, we introduce the concepts of s-pairwise MDS codes sets and s-pairwise MDS array codes sets. Two generic constructions (Generic Constructions I and II) for constructing the MSR code using the s-pairwise MDS codes sets or the s-pairwise MDS array codes sets are given. Constructions 1 to 3 proposed by Ye and Barg can be regarded as some special cases of Generic Construction I, and Constructions 1 to 3 proposed by Li et al., can be regarded as some special cases of Generic Construction II. It is worth mentioning that Generic Construction II can be applied to any finite field, including the binary field. We also demonstrate how to obtain the s-pairwise MDS code sets and the s-pairwise MDS array code sets from a high-rate MDS code or MDS array code over $\mathbb {F}_{q}$ . We obtain a novel class of MSR codes by utilizing the MDS array codes provided by Lv et al., as component codes according to Generic Construction II. As a byproduct of Constructions 4 to 7, we obtain a new class of MSR codes with optimal access property. Using two types of sets $\Gamma _{1}$ and $\Gamma _{2}$ with the property that matrices commute, we present several new constructions for MSR codes with the optimal access property over any finite field. In this paper, compared with the constructions over binary field proposed by Li et al., the sub-packetization of our constructions applicable to the binary field is significantly reduced.