We propose a unified construction for binary locally repairable codes (LRCs) with distance four. When there is a binary linear code $[n,k,4]$ without all zero coordinates, we can construct a binary LRC $[n,k,4]$ with very good locality r. Conditions for which these LRCs attain the Cadambe-Mazumdar bound are also presented. We prove that all our LRCs with locality $r\le 3$ are optimal LRCs, and most of our LRCs with locality $r\ge 4$ can achieve the Cadambe-Mazumdar bound. Especially, if $n\le 256$, all the constructed distance optimal $[n,k,4]$ LRCs, except 17 codes, can achieve the Cadambe-Mazumdar bound. We also show that six known constructions are covered in our construction, and eight known constructions can be improved by our construction.

我们为距离为4的二进制本地可修复代码（LRC）提出了统一的构造。有二进制线性代码时$[ñ，ķ，4]$ 没有所有零坐标，我们可以构造一个二进制LRC $[ñ，ķ，4]$有很好的地方r。还介绍了这些LRC达到Cadambe-Mazumdar界限的条件。我们证明我们所有本地的LRC$\mathrm{[R}\le 3$ 是最佳的LRC，而我们的大多数LRC都具有局部性 $\mathrm{[R}\ge 4$可以达到Cadambe-Mazumdar的界线。特别是如果$ñ\le 256$，所有构造的距离最佳 $[ñ，ķ，4]$除17个代码外，LRC可以达到Cadambe-Mazumdar边界。我们还表明，我们的结构涵盖了六个已知的构造，并且我们的构造可以改进八个已知的构造。