Bounds and Constructions of Singleton-Optimal Locally Repairable Codes With Small Localities

An $(n, k, d; r)_{q}$ -locally repairable code (LRC) is called a Singleton-optimal LRC if it achieves the Singleton-type bound. Analogous to the classical MDS conjecture, the maximal length problem of Singleton-optimal LRCs has attracted a lot of attention in recent years. In this paper, we give an improved upper bound for the length of q-ary Singleton-optimal LRCs with disjoint repair groups such that $(r+1)\mid n$ based on the parity-check matrix approach. In particular, for any Singleton-optimal $(n, k, d; r)_{q}$ -LRCs, we show that: 1) $n\le q+d-4$ , when $r=2$ and $d=3e+8$ with $e\ge 0$ ; 2) $n\leq (r+1)\left \lfloor {{\frac {2(q^{2}+q+1)}{r(r+1)} +e+1}}\right \rfloor $ , when $d\ge 8$ and $\max \left \{{{3,\frac {d-e-6}{e+1}}}\right \}\le r\le \frac {d-e-3}{e+1}$ for any $0\le e\le \left \lfloor {{\frac {d-6}{4} }}\right \rfloor $ . Furthermore, we establish equivalent connections between the existence of Singleton-optimal $(n,k,d;r)_{q}$ -LRCs for $d=6, r=3$ and $d=7, r=2$ with disjoint repair groups and some subsets of lines in finite projective space with certain properties. Consequently, we prove that the length of q-ary Singleton-optimal LRCs with minimum distance $d=6$ and locality $r=3$ is upper bounded by $O(q^{1.5})$ . We construct Singleton-optimal $(8\le n\le q+1,k,d=6,r=3)_{q}$ -LRC with disjoint repair groups such that $4\mid n$ and determine the exact value of the maximum code length for some specific q. We also prove the existence of $(n, k, d=7; r=2)_{q}$ -Singleton-optimal LRCs for $n \approx \sqrt {2}q$ .