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Binary maximum distance separable (MDS) array codes are a special class oferasure codes for distributed storage that not only provide fault tolerancewith minimum storage redundancy but also achieve low computational complexity.They are constructed by encoding $k$ information columns into $r$ paritycolumns, in which each element in a column is a bit, such that any $k$ out ofthe $k+r$ columns suffice to recover all information bits. In addition toproviding fault tolerance, it is critical to improve repair performance inpractical applications. Specifically, if a single column fails, our goal is tominimize the repair bandwidth by downloading the least amount of bits from $d$healthy columns, where $k\leq d\leq k+r-1$. If one column of an MDS code isfailed, it is known that we need to download at least $1/(d-k+1)$ fraction ofthe data stored in each of $d$ healthy columns. If this lower bound is achievedfor the repair of the failure column from accessing arbitrary $d$ healthycolumns, we say that the MDS code has optimal repair. However, if such lowerbound is only achieved by $d$ specific healthy columns, then we say the MDScode has weak-optimal repair. In this paper, we propose two explicitconstructions of binary MDS array codes with more parity columns (i.e., $r\geq3$) that achieve asymptotically weak-optimal repair, where $k+1\leq d\leqk+\lfloor(r-1)/2\rfloor$, and "asymptotic" means that the repair bandwidthachieves the minimum value asymptotically in $d$. Codes in the firstconstruction have odd number of parity columns and asymptotically weak-optimalrepair for any one information failure, while codes in the second constructionhave even number of parity columns and asymptotically weak-optimal repair forany one column failure.

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