Binary MDS Array Codes with Asymptotically Optimal Repair for All Columns

An m x (k+r) binary maximum distance separable (MDS) array code contains k information columns and r parity columns with each entry being a bit such that any k out of k+r columns can retrieve the k information columns. When there is a failed column, it is critical to minimize the repair bandwidth that is the total number of bits downloaded from d out of k+r-1 surviving columns in repairing the failed column. In this paper, we first propose a new construction of binary MDS array codes with any number of parity columns (i.e., r>= 2) that have asymptotically optimal repair bandwidth for any information column, where d=k+r-1. By applying a transformation for the proposed binary MDS array codes, we then can obtain the transformed binary MDS array codes that also have optimal repair bandwidth for any parity column.