Constructions of Optimal Binary Locally Repairable Codes With Multiple Repair Groups

In distributed storage systems, locally repairable codes (LRCs) have attracted lots of interest recently. If a code symbol can be repaired respectively by $t$ disjoint groups of other symbols, each of which has size at most $r$ , we say that the code symbol has $(r,t) $ -locality. LDPC codes are linear block codes defined by low-density parity-check matrices. A regular $(\tau,\rho)$ -LDPC code has the parity-check matrix with uniform column weight $\tau$ and uniform row weight $\rho$ . In this letter, we employ regular LDPC codes to construct optimal binary LRCs with $(r,t) $ -locality for information symbols. After proposing construction frameworks, three detailed constructions of binary LRCs with information locality are obtained, all of which have a single parity symbol in each repair group. All our codes attain the distance bounds of LRCs when each repair group contains a single parity symbol and thus are optimal. For storage systems with hot data, the proposed binary LRCs seem promising for system implementations since the encoding, repairing, parallel reading, and data reconstruction can be performed by simple XOR operations.