Generic Construction of Optimal-Access Binary MDS Array Codes with Smaller Sub-packetization

Abstract

A $(k+r,k,l)$ binary array code of length $k+r$, dimension $k$, and sub-packetization $l$ is composed of $l\times(k+r)$ matrices over $\mathbb{F}_2$, with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. It is said to be maximum distance separable (MDS) if any $k$ out of $k+r$ coordinates suffice to reconstruct the whole codeword. The repair problem of binary MDS array codes has drawn much attention, particularly for single-node failures. In this paper, given an arbitrary binary MDS array code with sub-packetization $m$ as the base code, we propose two generic approaches (Generic Construction I and II) for constructing binary MDS array codes with optimal access (or repair) bandwidth for single-node failures. For every $s\leq r$, a $(k+r,k,ms^{\lceil \frac{k+r}{s}\rceil})$ code $\mathcal{C}_1$ with optimal access bandwidth can be constructed by Generic Construction I. Repairing a failed node of $\mathcal{C}_1$ requires connecting to $d = k+s-1$ helper nodes, in which $s-1$ helper nodes are designated and $k$ are free to select. $\mathcal{C}_1$ generally achieves smaller sub-packetization and provides greater flexibility in the selection of its coefficient matrices. For even $r\geq4$ and $s=\frac{r}{2}$ such that $s+1$ divides $k+r$, a $(k+r, k,ms^{\frac{k+r}{s+1}})$ code $\mathcal{C}_2$ with optimal repair bandwidth can be constructed by Generic Construction II, with $\frac{s}{s+1}(k+r)$ out of $k+r$ nodes having the optimal access property. To the best of our knowledge, $\mathcal{C}_2$ possesses the smallest sub-packetization among existing binary MDS array codes with optimal repair bandwidth known to date.