短码长四元最优局部修复码的构造

在分布式存储系统中，当节点发生故障时局部修复码(LRC)可以通过访问少量其他节点来恢复数据，然而LRC的局部度不尽相同，该文构造了短码长且局部度较小的四元LRC。当码长不超过20，最小距离大于2时，若四元距离最优线性码的生成阵维数不超过校验阵维数，可利用其生成阵给出LRC，否则利用其校验阵给出LRC。对已构造的LRC的生成阵或校验阵，利用删除、并置等方法得到新矩阵，从而构造出190个码长$n le 20$，最小距离$d ge 2$的LRC。除12个LRC外，其他LRC是局部度最优的。

Constructions of Quaternary Optimal Locally Repairable Code with Short Length

In distributed storage system, when a node fails, Locally Repairable Code (LRC) can access other nodes to recover data. However, the locality of LRC is not the same. Quaternary LRC with short code length and small locality is constructed. When code length is not more than 20 and minimum distance is greater than 2, if the dimension of generator matrix of a quaternary distance optimal linear code does not exceed the dimension of parity-check matrix, an LRC can be constructed from generator matrix, otherwise parity-check matrix can be used to construct an LRC. From generator matrices or parity-check matrices of LRCs constructed, other LRC are given by operations of deleting and juxtaposition. There are 190 LRC with code length n ≤ 20 and minimum distance d ≥ 2 to be constructed. Except for 12 LRC, other LRC are all locality optimal.