A class of binary MDS array codes with asymptotically weak-optimal repair

Abstract

Binary maximum distance separable (MDS) array codes contain $k$ information columns and $r$ parity columns in which each entry is a bit that can tolerate $r$ arbitrary erasures. When a column in an MDS code fails, it has been proven that we must download at least half of the content from each helper column if $k+1$ columns are selected as the helper columns. If the lower bound is achieved such that the $k+1$ helper columns can be selected from any $k+r-1$ surviving columns, then the repair is an optimal repair. Otherwise, if the lower bound is achieved with $k+1$ specific helper columns, the repair is a weak-optimal repair. This paper proposes a class of binary MDS array codes with $k\geq~3$ and $r\geq~2$ that asymptotically achieve weak-optimal repair of an information column with $k+1$ helper columns. We show that there exist many encoding matrices such that the corresponding binary MDS array codes can asymptotically achieve weak-optimal repair for repairing any information column.