Improved bounds and Optimal Constructions of Locally Repairable Codes with distance 5 and 6

Repair locality has been an important metric in distributed storage systems (DSS). Erasure codes with small locality are more popular in DSS, which means fewer available nodes participating in the repair of failed nodes. Locally repairable codes (LRCs) peoposed as a new coding scheme, give more rise to the system performance and attract a lot of interest in the theoretical research in coding theory. The particular concern among the research problems is the bounds and optimal constructions of LRCs. In this direction, we first of all derive an improved upper bound on the code length of optimal LRCs with minimum distance d = 5, 6, some known constructions are shown to exactly achieve our new bound, which verifies its tightness. Then we construct a class of distance-optimal LRCs based on the structure of a sunflower, whose code length n = 3(q + 1), locality r = 2 and distance d = 6. Note that the code length of this class of LRCs outperforms all known optimal constructions with the same parameters. Moreover, by employing the combinatorial structure of the sunflower and the q-Steiner system, we obtain two classes of k-optimal (dimension-optimal) LRCs with respect to our new bound for d = 6. It’s worth noting that all of our optimal constructions possess small locality r = 2, which are attractive in DSS.