On optimal ternary locally repairable codes
Jie Hao, Shu‐Tao Xia, Bin Chen · 2017
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) locally repairable code (LRC), the minimum distance satisfies the well-known Singleton-like bound d ≤ n - k - [k/r] + 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.