Improved Bounds and Singleton-Optimal Constructions of Locally Repairable Codes With Minimum Distance 5 and 6

Repair locality has been an important metric in a distributed storage system (DSS). Erasure codes with small locality are more popular in a DSS, which means fewer available nodes participating in the repair process of failed nodes. Locally repairable codes (LRCs) as a new coding scheme have given more rise to the system performance and attracted 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. The problem of optimal constructions of LRCs includes the most important case of Singleton-optimal LRCs whose minimum distance achieves the Singleton-like bound, which is the core consideration in this paper. In this work, we first of all derive an improved and general upper bound on the code length of Singleton-optimal LRCs with minimum distance d = 5, 6, some known constructions are shown to exactly achieve our new bound, which verifies its tightness. For locality r = 2 and distance d = 6, we construct three newSingleton-optimal LRCs whose code length n = 3(q + 1), n = 3(q + √q + 1) and n = 3(2q - 4), respectively. Moreover, we obtain a complete characterization for Singletonoptimal LRCs with r = 2 and d = 6. Such characterization has established an important connection between the existence of Singleton-optimal LRCs and that of a special subset of lines of finite projective plane P G(2, q), thus provides a methodology for constructing LRCs with longer length based on any advance on finite projective plane P G(2, q). In the end, we employ the well-known line-point incidence matrix and Johnson bounds for constant weight codes to derive tighter upper bounds on the code length. These new bounds further help us to prove that some of the previous Singleton-optimal constructions or their extensions achieve the longest possible code length for q = 3, 4, 5, 7. It's worth noting that all of our Singleton-optimal constructions possess small locality r = 2, which are attractive in a DSS.