Constructions of Binary Optimal Locally Repairable Codes via Intersection Subspaces

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 the binary linear LRCs can significantly reduce the coding and decoding complexity, the construction of binary LRCs is of particular interest. To date, all the known optimal binary linear LRCs with the locality $2^b$ ($b\geq 3$) are based on the so-called partial spread which is a collection of the same dimensional subspaces with pairwise trivial, i.e., zero-dimensional intersection. In this paper, we concentrate on binary linear LRCs with disjoint local repair groups. We 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 vs. the traditional partial spread construction. This method will increase the number of possible repair groups of LRCs as many as possible, and thus efficiently enlarge 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 $2^b$ ($b\geq 3$).