  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 $2^b$ ($b\geq 3$).
