Some New Results on Improved Bounds and Constructions of Singleton-Optimal (r,δ) Locally Repairable Codes

In this paper, we focus on Singleton-optimal $(r,\delta)$ LRCs with disjoint local repair groups. We provide an improved bound for the length of q-ary Singleton-optimal $(r,\delta)$ LRCs based on the parity-check matrix approach. Specifically, for $d \geq 3\delta $ , we prove that $n\le O(q^{\delta })$ when $d-3\delta \lt r\le d-2\delta +1$ . We also show that the code length $n\le q+\delta +2$ when $r=2$ and $d=3\delta +2$ . We present a sufficient and necessary condition for the existence of Singleton-optimal $(n,k,d;r,\delta)$ LRCs with disjoint local repair groups, where the minimum distance satisfies $3\delta +1\le d \le 3\delta +2$ and locality $r=2$ . This condition imposes an upper bound on the code length, $n\le O(q^{2})$ , and indicates the existence of a code length approximately given by $n\approx \sqrt {2}q$ when $d=3\delta +1$ and $r=2$ . Finally, we utilize blocking sets to provide a general construction of Singleton-optimal $(n,k,d=2\delta +2,r=2,\delta)$ LRC with code length $n\approx O\left ({{q^{\frac {h+1}{h}}}}\right)$ for any $h\ge 3$ . To the best of our knowledge, this is the first family of Singleton-optimal $(n,k,d=2\delta +2,r=2,\delta)$ LRC with super-linear code length.