Some New Results on Improved Bounds and Constructions of Singleton-Optimal (r,δ) Locally Repairable Codes
Ran Tao, Weijun Fang, Ye Wang, Fang‐Wei Fu, Sihuang Hu · IEEE Transactions on Communications · 2024
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.