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.

翻译：摘要：最小存储再生（MSR）码具有MDS特性和最优修复带宽，广泛应用于分布式存储系统（DSS）的数据恢复。本文考虑集中式模型下$(n,k,l)$ MSR码的构造，该码可同时修复$h$个故障节点，且$d$个辅助节点中有$e$个提供错误信息。我们首先提出新的修复方案，并完整证明当部分辅助节点提供错误信息时，从辅助节点下载符号量的下界。随后重点研究基于该修复方案的两种显式构造。对于$2\leq h\leq n-k$，$k+2e\leq d \leq n-h$且$d\equiv k+2e \;(\mod{h})$，第一种构造具有UER $(h, d)$-最优修复特性，第二种构造具有UER $(h, d)$-最优访问特性。与原始构造（Ye和Barg，IEEE Tran. Inf. Theory，第63卷，2017年4月）相比，我们的构造在三个方面有所改进：1）所提修复方案比Ye和Barg提出的逐节点方案在并行数据系统中更具可行性；2）子包化程度从$\left(\operatorname{lcm}(d-k+1, d-k+2,\cdots, d-k+h)\right)^n$降低至$\left((d-2e-k+h)/h\right)^n，至少降低$(h(d-k+h))^n$倍；3）第一种构造的域大小降低至$|\mathbb{F}| \geq n(d-2e-k+h)/h\)，至少降低$h(d-k+h)$倍。在实际应用中，由于存储容量有限且编码、解码和修复过程的计算复杂度要求较低，较小的子包化程度和域大小更具优势。