Binary MDS Array Codes with Optimal Repair

Consider a binary maximum distance separable (MDS) array code composed of an $m\times (k+r)$ array of bits with $k$ information columns and $r$ parity columns, such that any $k$ out of $k+r$ columns suffice to reconstruct the $k$ information columns. Our goal is to provide {\em optimal repair access} for binary MDS array codes, meaning that the bandwidth triggered to repair any single failed information or parity column is minimized. In this paper, we propose a generic transformation framework for binary MDS array codes, using EVENODD codes as a motivating example, to support optimal repair access for $k+1\le d \le k+r-1$, where $d$ denotes the number of non-failed columns that are connected for repair; note that when $d<k+r-1$, some of the chosen $d$ columns in repairing a failed column are specific. In addition, we show how our transformation framework applies to an example of binary MDS array codes with asymptotically optimal repair access of any single information column and enables asymptotically or exactly optimal repair access for any column. Furthermore, we present a new transformation for EVENODD codes with two parity columns such that the existing efficient repair property of any information column is preserved and the repair access of parity column is optimal.