Locally repairable codes (LRCs) are a family of erasure codes which have been proposed for data storage in distributed storage systems recently. For an LRC, a code symbol is said to have locality r and availability t, or (r, t)-locality, if the code symbol can be repaired respectively by t disjoint groups of other symbols, each of which has size at most r. For [n, k, d] LRCs with length n, dimension k, minimum distance d and (r, t)-locality for information symbols, and each repair group contains a single parity symbol, Rawat et al. derived a Singleton-like bound: d ≤ n − k − ⌈kt/r⌉ + t + 1. In this paper, we conduct an in-depth study of this bound based on a parity-check matrix approach. An alternative simple proof of this bound is given, and necessary conditions for optimal LRCs attaining this bound are obtained. It is shown that the previous optimal codes attaining this bound satisfy the necessary condition. Furthermore, two classes of optimal binary LRCs attaining this bound are proposed under the guideline of this necessary condition.
['Jie Hao', 'Shu-Tao Xia', 'Bin Chen']
On the single-parity locally repairable codes with availability