低维四元局部修复码的构造

Construction of Quaternary Locally Repairable Code with Low Dimension

在分布式存储系统中,当节点发生故障时,局部修复码能够提高修复效率。四元距离最优码易于实现,当给定码长和维数时,四元距离最优码的纠错能力优于二元距离最优码,但目前利用四元距离最优码构造四元局部修复码的研究存在很多空白。设四元距离最优码的维数2≤k≤4,由给定维数的四元Simplex码与MacDonald码以及少量距离最优码的生成矩阵,利用扩展、删除与并置等组合方法,设法构造出任意码长n≥k+1且局部度较小的四元局部修复码。确定出达到Singleton-Like界或Cadambe-Mazumdar界的四元局部修复码。证明除55个四元局部修复码外,其余的四元局部修复码都是局部度最优的。

In the distributed storage system,locally repairable code can improve repair efficiency when a node is at fault.Quaternary distance optimal codes are easy to realize and their error-correcting ability is superior to that of the binary distance optimal codes when code length and dimension are given.However,there are lots of work needed to construct quaternary locally repairable codes by quaternary distance optimal codes.When the dimensions of quaternary distance optimal codes are 2≤k≤4,by combination with the operation of combination,such as extension,the deletion and the juxtaposition,the paper can obtain generator matrices of quaternary Simplex code,MacDonald code and few distance optimal linear codes to construct quaternary locally repairable codes with code length n≥k+1 and small locality,and can verify quaternary locally repairable codes to attain Singleton-Like bound or Cadambe-Mazumdar bound and prove that except for 55 quaternary locally repairable codes,other quaternary locally repairable codes are all locality optimal.