On Codes for Optimal Rebuilding Access
  MDS (maximum distance separable) array codes are widely used in storage systems due to their computationally efficient encoding and decoding procedures. An MDS code with r redundancy nodes can correct any r erasures by accessing (reading) all the remaining information in both the systematic nodes and the parity (redundancy) nodes. However, in practice, a single erasure is the most likely failure event; hence, a natural question is how much information do we need to access in order to rebuild a single storage node? We define the rebuilding ratio as the fraction of remaining information accessed during the rebuilding of a single erasure. In our previous work we showed that the optimal rebuilding ratio of 1/r is achievable (using our newly constructed array codes) for the rebuilding of any systematic node, however, all the information needs to be accessed for the rebuilding of the parity nodes. Namely, constructing array codes with a rebuilding ratio of 1/r was left as an open problem. In this paper, we solve this open problem and present array codes that achieve the lower bound of 1/r for rebuilding any single systematic or parity node. 
Zhiying Wang, Itzhak Tamo, Jehoshua Bruck
cs.IT cs.DC math.IT