Locally repairable codes (LRCs), which can recover any symbol of a codeword by reading only a small number of other symbols, have been widely used in real-world distributed storage systems, such as Microsoft Azure Storage and Ceph Storage Cluster. Since binary linear LRCs can significantly reduce coding and decoding complexity, constructions of binary LRCs are of particular interest. The aim of this paper is to construct dimensional optimal binary locally repairable codes with disjoint local repair groups. We introduce how to connect intersection subspaces with binary locally repairable codes and construct dimensional optimal binary linear LRCs with locality $2^b$ ($b\geq 3$) and minimum distance $d\geq 6$ by employing intersection subspaces deduced from the direct sum. This method will sufficiently increase the number of possible repair groups of dimensional optimal LRCs, and thus efficiently expanding the range of the construction parameters while keeping the largest code rates compared with all known binary linear LRCs with minimum distance $d\geq 6$ and locality $2^b$ ($b\geq 3$).

翻译：局部修复码（LRCs）能够通过仅读取少量其他符号来恢复码字的任意符号，已广泛应用于实际分布式存储系统，如微软Azure存储和Ceph存储集群。由于二元线性LRCs能显著降低编解码复杂度，二元LRCs的构造具有特殊研究价值。本文旨在构造具有不相交局部修复组的维数最优二元局部修复码。我们引入如何将交子空间与二元局部修复码关联，并通过利用由直和导出的交子空间构造局部性为$2^b$（$b\geq 3$）、最小距离$d\geq 6$的维数最优二元线性LRCs。该方法能充分增加维数最优LRCs的可能修复组数量，从而在保持与已知最小距离$d\geq 6$、局部性$2^b$（$b\geq 3$）的二元线性LRCs相比最大码率的同时，有效扩展构造参数范围。