New Constructions of MDS Array Codes and Optimal Locally Repairable Array Codes

MDS array codes have been extensively studied due to their applications in storage systems. In this paper, we first propose a novel method of constructing MDS array codes by deleting one row and one column from the circulant matrices associated to some polynomials. Several new classes of MDS array codes with flexible parameters are constructed. In particular, we give a new algebraic presentation of the Blaum-Roth codes with sparser parity-check matrices. We also obtain a family of MDS array codes over finite fields with even characteristics whose parity-check matrices have the lowest density. Furthermore, based on these new MDS array codes, we give a general construction of optimal locally repairable array codes (LRACs) achieving the Singleton-type bound. Additionally, we obtain some new optimal LRACs of long lengths. Finally, we present a scheduled algorithm for syndrome computations of binary optimal LRACs with redundancy 4, which can tolerate three failures. The number of XORs per data bit required in our algorithm approaches 2 as the length approaches infinity, which is the same as the MDS codes tolerating three failures. However, the number of nodes required during the repair of a failed node in our optimal LRACs is only about half of that in MDS array codes.