Bounds and Constructions of Singleton-Optimal Locally Repairable Codes With Small Localities
Weijun Fang, Ran Tao, Fang‐Wei Fu, Bin Chen, Shu‐Tao Xia · IEEE Transactions on Information Theory · 2024
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$n \approx \sqrt {2}q$.