Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$
Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$
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DOI：
10.1109/access.2019.2934769
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发表时间：
2019
期刊：
影响因子：
3.9
通讯作者：
Yichong Xia;Bin Chen
中科院分区：
文献类型：
--
作者：
Yichong Xia;Bin Chen
A locally repairable code (LRC) is a $[n, k, d]$ linear code with length $n$ , dimension $k$ , minimum distance $d$ and locality $r$ , which means that every code symbol can be repaired by at most $r$ other symbols. LRCs have become an important candidate in distributed storage systems due to their relatively low I/O cost. An LRC is said to be optimal if its minimum distance meets one of the Singleton-like bounds. This paper considers the optimal constructions of LRCs with locality $r=1$ and $r=k-1$ , which involves three types: $r$ -local LRCs, $(r,\delta)$ -LRCs and LRCs with $t$ -availability. Specifically, we first prove that the existence of an optimal LRC with locality $r=1$ is equivalent to that of an MDS code with certain parameters. Thus we can completely characterize the three types of optimal LRCs with $r=1$ based on some known constructions of MDS codes. Near MDS codes is a special class of sub-optimal linear code whose minimum distance $d=n-k$ and the $i$ -th generalized Hamming weight achieves the generalized Singleton bound for $2\leq i \leq k$ . For $r=k-1$ , we have established the connections between optimal $r$ -local LRCs/ LRCs with $t$ -availability and near MDS codes. Such connections can help to construct optimal LRCs with $r=k-1$ from some known classes of near MDS codes.
