A New Centralized Multi-Node Repair Scheme of MSR codes with Error-Correcting Capability

Minimum storage regenerating (MSR) codes, with the MDS property and the optimal repair bandwidth, are widely used in distributed storage systems (DSS) for data recovery. In this paper, we consider the construction of $(n,k,l)$ MSR codes in the centralized model that can repair $h$ failed nodes simultaneously with $e$ out $d$ helper nodes providing erroneous information. We first propose the new repair scheme, and give a complete proof of the lower bound on the amount of symbols downloaded from the helped nodes, provided that some of helper nodes provide erroneous information. Then we focus on two explicit constructions with the repair scheme proposed. For $2\leq h\leq n-k$, $k+2e\leq d \leq n-h$ and $d\equiv k+2e \;(\mod{h})$, the first one has the UER $(h, d)$-optimal repair property, and the second one has the UER $(h, d)$-optimal access property. Compared with the original constructions (Ye and Barg, IEEE Tran. Inf. Theory, Vol. 63, April 2017), our constructions have improvements in three aspects: 1) The proposed repair scheme is more feasible than the one-by-one scheme presented by Ye and Barg in a parallel data system; 2) The sub-packetization is reduced from $\left(\operatorname{lcm}(d-k+1, d-k+2,\cdots, d-k+h)\right)^n$ to $\left((d-2e-k+h)/h\right)^n$, which reduces at least by a factor of $(h(d-k+h))^n$; 3) The field size of the first construction is reduced to $|\mathbb{F}| \geq n(d-2e-k+h)/h$, which reduces at least by a factor of $h(d-k+h)$. Small sub-packetization and small field size are preferred in practice due to the limited storage capacity and low computation complexity in the process of encoding, decoding and repairing.