A Generalization of Array Codes with Local Properties and Efficient Encoding/Decoding

A maximum distance separable (MDS) array code is composed of m× (k+ r) arrays such that any k out of k + r columns suffice to retrieve all the information symbols. Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity (EIP) codes are two classes of MDS array codes that can repair any one symbol in a column by locally accessing some other symbols within the column, where the number of symbols m in a column is a prime number. By generalizing the constructions of EBR and EIP codes, we propose new MDS array codes, such that any one symbol can be locally recovered and the number of symbols in a column can be not only a prime number but also a power of an odd prime number. Also, we present an efficient encoding/decoding method for the proposed generalized EBR (GEBR) and generalized EIP (GEIP) codes based on the LU factorization of a Vandermonde matrix. We show that the proposed decoding method has less computational complexity than existing methods. Furthermore, we show that the proposed GEBR codes have both a larger minimum symbol distance and a larger recovery ability of erased lines for some parameters when compared to EBR codes. We show that EBR codes can recover any r erased lines of a slope for any parameter r, which was an open problem in [2]. This paper was presented in part at the IEEE Global Communications Conference (GLOBECOM), 2020 [1]. H. Hou is with the School of Electrical Engineering & Intelligentization, Dongguan University of Technology (E-mail: houhanxu@163.com). Y. S. Han is with the Shenzhen Institute for Advanced Study, University of Electronic Science and Technology of China (E-mail: yunghsiang@gmail.com). P. P. C. Lee is with Department of Computer Science and Engineering, The Chinese University of Hong Kong (E-mail: pclee@cse.edu.cuhk.hk). Y. Wu and G. Han are with the School of Information Engineering, Guangdong University of Technology (E-mail: 278008313@qq.com, gjhan2000@gmail.com). M. Blaum is with IBM Research DivisionAlmaden (E-mail: mblaum@hotmail.com). This work was partially supported by the National Key R&D Program of China (No. 2020YFA0712300) and the National Natural Science Foundation of China (No. 62071121), Research Grants Council of HKSAR (AoE/P-404/18) and Innovation and Technology Fund (ITS/315/18FX). October 12, 2021 DRAFT 2 Index Terms Array codes, Expanded-Blaum-Roth codes, Expanded-Independent-Parity codes, local repair, efficient encoding/decoding.

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