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

TL;DR

This paper introduces two generic methods for constructing binary MDS array codes with optimal access and repair bandwidth, achieving smaller sub-packetization and greater flexibility, especially for certain parameters, advancing the efficiency of distributed storage systems.

Contribution

It proposes two new generic constructions for binary MDS array codes with optimal repair properties, reducing sub-packetization and enhancing flexibility over existing codes.

Findings

Construction $$ achieves smaller sub-packetization.

Construction $$ attains the smallest known sub-packetization for optimal repair.

Codes provide flexible helper node selection for repair.

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.