Construction of Locally Repairable Codes with Multiple Localities Based on Encoding Polynomial

Locally repairable codes, which can repair erased symbols from other symbols, have attracted a good deal of attention in recent years because its local repair property is effective on distributed storage systems. (r(u), delta(u))(u is an element of[s])(- locally) repairable codes with multiple localities, which are an extension of ordinary locally repairable codes, can repair delta(u) - 1 erased symbols simultaneously from a set consisting of at most r(u) symbols. An upper bound on the minimum distance of these codes and a construction method of optimal codes, attaining this bound with equality, were given by Chen, Hao, and Xia. In this paper, we discuss the parameter restrictions of the existing construction, and we propose explicit constructions of optimal codes with multiple localities with relaxed restrictions based on the encoding polynomial introduced by Tamo and Barg. The proposed construction can design a code whose minimum distance is unrealizable by the existing construction.