Constructions of locally repairable codes via concatenated codes

TL;DR

This paper presents a systematic construction of optimal binary locally repairable codes (LRCs) using concatenated codes with outer codes over 4, achieving bounds and improving existing theoretical limits.

Contribution

The authors introduce a new method for constructing binary LRCs via concatenated codes, including optimal and perfect codes, and improve bounds for specific locality cases.

Findings

Constructed optimal binary LRCs meeting the Griesmer-like bound.

Developed a systematic approach using 4-linear outer codes.

Improved the Johnson-like bound for locality r=2 LRCs.

Abstract

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.