Optimal Locally Repairable Codes With Multiple Erasure Tolerance
Jing Wang, Yan Ma, Tingting Yu, Kun Yang, Xiangyang Liu · IEEE Communications Letters · 2025
To address the issues of stringent parameter constraints and lower code rates in the construction of locally repairable codes (LRCs) under multiple erasures, this letter proposes (r, δ,t)-LRCs with flexible availability and hierarchical LRCs (H-LRCs) with flexible levels, based on resolvable group divisible design (RGDD). Specifically, using cyclic planar difference sets to obtain the triple of RGDD, LRCs with (r, δ,t)-information-locality are constructed based on the parallel class blocks of RGDD. Furthermore, H-LRCs are designed for information symbols with parameters ( (r1, δ1), (r2, δ2), . . . , (rh, δh) ) by combining the elements within RGDD groups. Both the constructed LRCs and H-LRCs achieve optimal minimum distances. Compared to existing (r, δ,t)-LRCs, the proposed (r, δ,t)-LRCs provide more flexible availability, fewer parameter constraints, and higher code rates. In contrast to current H-LRCs, the proposed H-LRCs impose fewer parameter constraints and allow the number of levels to be adjusted according to the number of information nodes. Moreover, the proposed H-LRCs optimize the parameters of (r, δ,t)-LRCs, resulting in higher code rates.