A New Design of Binary MDS Array Codes with Asymptotically Weak-Optimal Repair

Binary maximum distance separable (MDS) array codes are a special class of erasure codes for distributed storage that not only provide fault tolerance with minimum storage redundancy but also achieve low computational complexity. They are constructed by encoding k information columns into r parity columns, in which each element in a column is a bit, such that any k out of the k+r columns suffice to recover all information bits. In addition to providing fault tolerance, it is critical to improve repair performance in practical applications. Specifically, if a single column fails, our goal is to minimize the repair bandwidth by downloading the least amount of bits from d healthy columns, where k≤ d≤ k+r-1. If one column of an MDS code is failed, it is known that we need to download at least 1/(d-k+1) fraction of the data stored in each of d healthy columns. If this lower bound is achieved for the repair of the failure column from accessing arbitrary d healthy columns, we say that the MDS code has optimal repair. However, if such lower bound is only achieved by d specific healthy columns, then we say the MDS code has weak-optimal repair. In this paper, we propose two explicit constructions of binary MDS array codes with more parity columns (i.e., r≥ 3) that achieve asymptotically weak-optimal repair, where k+1≤ d≤ k+⌊(r-1)/2⌋, and "asymptotic" means that the repair bandwidth achieves the minimum value asymptotically in d. Codes in the first construction have odd number of parity columns and asymptotically weak-optimal repair for any one information failure, while codes in the second construction have even number of parity columns and asymptotically weak-optimal repair for any one column failure.