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Recently, Hao and Xia noted a connection between a class of binary locally repairable codes (LRCs) with multiple repair groups and binary low-density parity-check (LDPC) codes, and proposed a framework for constructing binary LRCs from LDPC codes as well as three specific constructions of binary LRCs that can achieve a distance bound. The connection between binary LRCs and LDPC codes, however, has not yet been fully disclosed and constructions of binary LRCs that can achieve the distance bound remain largely unknown. Accordingly, this paper comments on the connection and presents two infinite families of binary LRCs that can achieve the distance bound, based on circulant permutation matrices and affine permutation matrices. The proposed LRCs generalize the promising construction of high-rate codes proposed by Hao and Xia, and offer larger relative distances with the same or higher code rates when compared with other competitive codes.

On Constructions of a Class of Binary Locally Repairable Codes With Multiple Repair Groups