Constructions of locally repairable codes via concatenated codes
For distributed storage systems, this work provides a systematic method to construct optimal binary LRCs, improving theoretical bounds and offering explicit code constructions.
The paper constructs optimal binary locally repairable codes (LRCs) via concatenated codes, achieving codes that meet the Griesmer-like bound and perfect LRCs, and improves the Johnson-like bound for binary LRCs with locality 2, with explicit constructions attaining the new bound.
In recent years, locally repairable codes (LRCs) have attracted considerable attention owing to their pivotal role in distributed storage systems. Since binary linear locally repairable codes can significantly reduce the complexity of both encoding and decoding processes, the construction of binary LRCs has attracted extensive research interest. In this paper, we construct locally repairable codes via concatenated codes and present a systematic approach to select outer codes to obtain optimal binary LRCs, where the outer codes are linear codes over $\mathbb{F}_4$. The weight distributions of the resulting LRCs are determined by the weight distributions of the selected linear codes over $\mathbb{F}_4$. Furthermore, several classes of optimal binary locally repairable codes are constructed, including binary LRCs meeting the Griesmer-like bound, and binary perfect LRCs. Meanwhile, for the locality $r=2$, we improve the Johnson-like bound for binary LRCs with disjoint local repair groups established by Ma and Ge, and construct explicit LRCs that attain this new bound.