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Paper Title Classification of Optimal Ternary (r,\delta)-Locally Repairable Codes Attaining the Singleton-like Bound
Paper IdentifierFR3.R3.3
Authors Jie Hao, Beijing University of Posts and Telecommunications, China; Kenneth W. Shum, The Chinese University of Hong Kong, Shenzhen, China; Shu-Tao Xia, Tsinghua University, China; Yi-Xian Yang, Beijing University of Posts and Telecommunications, China
Session Locally Repairable Codes
Location Monge, Level 3
Session Time Friday, 12 July, 14:30 - 16:10
Presentation Time Friday, 12 July, 15:10 - 15:30
Manuscript  Click here to download the manuscript
Abstract In a linear code, a code symbol with (r,\delta)-locality can be repaired by accessing at most r other code symbols in case of at most \delta-1 erasures. A q-ary (n,k,r,\delta) locally repairable codes (LRC) in which every code symbol has (r,\delta)-locality is said to be optimal if it achieves the Singleton-like bound derived by Prakash et al.. In this paper, we study the classification of optimal ternary (n,k,r,\delta)-LRCs (\delta > 2). Firstly, we propose an upper bound on the minimum distance of optimal q-ary LRCs in terms of the field size. Then, we completely determine all the 6 classes of possible parameters with which optimal ternary (n,k,r,\delta)-LRCs exist. Moreover, explicit constructions of all these 6 classes of optimal ternary LRCs are proposed in the paper.