|Weight Distributions of q-ary Optimal Locally Repairable Codes with Locality 2, Distance 5 and Even Dimension
|Jie Hao, Beijing University of Posts and Telecommunications, China; Jun Zhang, Capital Normal University, China; Shu-Tao Xia, Tsinghua University, China; Fang-Wei Fu, Nankai University, China; Yi-Xian Yang, Beijing University of Posts and Telecommunications, China
|M.3: Codes for Distributed Storage III
|Coding for Storage and Memories
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|The weight distribution of a q-ary [n,k,d] linear code is an important research subject in coding theory. In a linear code, a code symbol is said to have locality r if it can be recovered by accessing at most r other code symbols. A q-ary locally repairable code (LRC) is an [n,k,d] linear code over F_q such that every code symbol has locality r, and is said to be optimal if the minimum distance attains the well-known Singleton-like bound. In this paper, we focus on the weight distributions of q-ary optimal LRCs with locality 2, minimum distance 5 and even dimension k. By analyzing the parity-check matrices involving locality, it is shown that the weight distributions of all q-ary optimal LRCs with locality 2, distance 5, even dimension k and code length n can be uniquely determined and explicit expressions of the weight distributions are given.