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Minimum storage regenerating (MSR) codes, with the MDS property and theoptimal repair bandwidth, are widely used in distributed storage systems (DSS)for data recovery. In this paper, we consider the construction of $(n,k,l)$ MSRcodes in the centralized model that can repair $h$ failed nodes simultaneouslywith $e$ out $d$ helper nodes providing erroneous information. We first proposethe new repair scheme, and give a complete proof of the lower bound on theamount of symbols downloaded from the helped nodes, provided that some ofhelper nodes provide erroneous information. Then we focus on two explicitconstructions with the repair scheme proposed. For $2\leq h\leq n-k$, $k+2e\leqd \leq n-h$ 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 $(h, d)$-optimalaccess property. Compared with the original constructions (Ye and Barg, IEEETran. Inf. Theory, Vol. 63, April 2017), our constructions have improvements inthree aspects: 1) The proposed repair scheme is more feasible than theone-by-one scheme presented by Ye and Barg in a parallel data system; 2) Thesub-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 reducesat least by a factor of $(h(d-k+h))^n$; 3) The field size of the firstconstruction is reduced to $|\mathbb{F}| \geq n(d-2e-k+h)/h$, which reduces atleast by a factor of $h(d-k+h)$. Small sub-packetization and small field sizeare preferred in practice due to the limited storage capacity and lowcomputation complexity in the process of encoding, decoding and repairing.

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