A New Design of Binary MDS Array Codes with Asymptotically Weak-Optimal Repair

Binary maximum distance separable (MDS) array codes are a special class of erasure codes for distributed storage that not only provide fault tolerance with minimum storage redundancy but also achieve low computational complexity. They are constructed by encoding k information columns into r parity columns, in which each element in a column is a bit, such that any k out of the k + r columns suffice to recover all information bits. In addition to providing fault tolerance, it is critical to improve repair performance in practical applications. Specifically, if a single column fails, our goal is to minimize the repair bandwidth by downloading the least amount of bits from d healthy columns, where k ≤ d ≤ k + r − 1. If one column of an MDS code is failed, it is known that we need to download at least 1/(d− k + 1) fraction of the data stored in each of d healthy columns. If this lower bound is achieved for the repair of the failure column from accessing arbitrary d healthy columns, we say that the MDS code has optimal repair. However, if such lower bound is only achieved by d specific healthy columns, then we say the MDS code has weak-optimal repair. Existing binary MDS array codes that achieve high data rate (i.e., k/(k + r) > 1/2) and optimal repair of information column only support double fault tolerance (i.e., r = 2), which is insufficient for failure-prone distributed storage environments in practice. This paper fills the void by proposing two explicit constructions of binary MDS array codes with more parity columns (i.e., r ≥ 3) that achieve asymptotically weak-optimal repair, where k + 1 ≤ d ≤ k + b(r − 1)/2c. Codes in the first construction have odd number of parity This paper was presented in part in [1] at the IEEE International Symposium on Information Theory, Aachen, Germany, June 2017. H. Hou is with the School of Electrical Engineering & Intelligentization, Dongguan University of Technology and the Shenzhen Graduate School, Peking University (E-mail: [email protected]). Y. S. Han is with the School of Electrical Engineering & Intelligentization, Dongguan University of Technology (E-mail: [email protected]). P. P. C. Lee is with Department of Computer Science and Engineering, The Chinese University of Hong Kong (E-mail: [email protected]). Y. Hu is with the School of Computer Science and Technology, Huazhong University of Science and Technology (E-mail: [email protected]). H. Li is with the Shenzhen Graduate School, Peking University (E-mail: [email protected]). This work was partially supported by the National Natural Science Foundation of China (No. 61701115, 61671007, 61671001) and National Keystone R&D Program of China (No. 2017YFB0803204, 2016YFB0800101). February 28, 2018 DRAFT ar X iv :1 80 2. 07 89 1v 2 [ cs .I T ] 2 7 Fe b 20 18 2 IEEE TRANSACTIONS ON INFORMATION THEORY columns and asymptotically weak-optimal repair for any one information failure, while codes in the second construction have even number of parity columns and asymptotically weak-optimal repair for any one column failure.