Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups
Jingxue Ma, Gennian Ge · SIAM Journal on Discrete Mathematics · 2019
In recent years, several classes of codes have been introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which leads to the systems being computationally expensive. Due to this, binary LRCs are of interest in practice. In this paper, we focus on binary linear LRCs with disjoint repair groups. We first derive an explicit bound for the dimension $k$ of such codes, which can be viewed as a generalization of the bounds given in [S. Goparaju and R. Calderbank, Proceedings of the IEEE International Symposium on Information Theory, 2014, pp. 676--680; A. Wang, Z. Zhang, and D. Lin, Proceedings of the IEEE International Symposium on Information Theory, 2017, pp. 2033--2037; and A. Zeh and E. Yaakobi, Proceedings of the IEEE Information Theory Workshop, 2015, pp. 1--5]. We also give several new constructions of binary LRCs with minimum distance $d=6$ based on weakly independent sets and partial spreads, which are optimal with respect to our bound. In particular, for locality $r\in \{2,3\}$ and minimum distance $d=6$, we construct optimal binary linear LRCs with disjoint repair groups for almost all possible parameters. It should be emphasized that some subclasses of these newly constructed codes also attain a general bound of binary LRCs (without the assumption of disjoint repair groups).