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arXiv · 1702.05730

On Optimal Ternary Locally Repairable Codes

Jie Hao·Shu-Tao Xia·Bin Chen
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Abstract

In an $[n,k,d]$ linear code, a code symbol is said to have locality $r$ if it can be repaired by accessing at most $r$ other code symbols. For an $(n,k,r)$ \emph{locally repairable code} (LRC), the minimum distance satisfies the well-known Singleton-like bound $d\le n-k-\lceil k/r\rceil +2$. In this paper, we study optimal ternary LRCs meeting this Singleton-like bound by employing a parity-check matrix approach. It is proved that there are only $8$ classes of possible parameters with which optimal ternary LRCs exist. Moreover, we obtain explicit constructions of optimal ternary LRCs for all these $8$ classes of parameters, where the minimum distance could only be 2, 3, 4, 5 and 6.

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Jie Hao, Shu-Tao Xia, Bin Chen. 2017-02-19. On Optimal Ternary Locally Repairable Codes. https://arxiv.org/abs/1702.05730

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