Constructing MDS Array Codes with Small Repair Bandwidth under Sub-packetization Two.

Erasure Codes are widely implemented in distributed storage systems to achieve fault tolerance with high storage efficiency. Reed-Solomon code is commonly deployed in data centers due to its optimal storage efficiency, but it requires massive repair bandwidth for node repair. Minimum Storage Regenerating (MSR) code and Locally Repairable (LR) code are proposed to reduce repair bandwidth. However, MSR code usually carries a heavy disk I/O burden and LR code is not optimal in storage efficiency. In this paper, we take disk I/O, storage efficiency, repair bandwidth, sub-packetization level, and repair degree (number of helper nodes) into consideration together, and propose explicit construction of MDS array code (named LDIO code) with low disk I/O and reduced repair bandwidth, where the sub-packetization level is 2. Besides the low disk I/O consumption, LDIO code also achieves a small repair bandwidth of ${\bar \gamma _{sys}} = \frac{1}{2} + \frac{1}{{4(r - 1)}}$ for systematic nodes with repair degree d = k+1. Normally, LDIO code can provide bandwidth savings of 25% to 50% for the repair of systematic nodes compared with RS code, and almost savings of 6% compared with piggybacking code. For an LDIO code of length n and redundancy r, the size of the finite field needed is Θ(n ^r+1 ).