Constructions of Binary Optimal Locally Repairable Codes via Intersection Subspaces

TL;DR

This paper introduces a novel method using intersection subspaces to construct binary locally repairable codes with optimal dimensions, high code rates, and specific parameters suitable for distributed storage systems.

Contribution

It presents a new construction technique for binary LRCs with disjoint repair groups using intersection subspaces, expanding the range of parameters while maintaining optimality.

Findings

Constructed binary LRCs with locality 2^b and minimum distance ≥6.

Increased the number of repair groups compared to existing codes.

Achieved high code rates while maintaining optimality.

Abstract

Locally repairable codes (LRCs), which can recover any symbol of a codeword by reading only a small number of other symbols, have been widely used in real-world distributed storage systems, such as Microsoft Azure Storage and Ceph Storage Cluster. Since binary linear LRCs can significantly reduce coding and decoding complexity, constructions of binary LRCs are of particular interest. The aim of this paper is to construct dimensional optimal binary locally repairable codes with disjoint local repair groups. We introduce how to connect intersection subspaces with binary locally repairable codes and construct dimensional optimal binary linear LRCs with locality $2^b$ ($b\geq 3$) and minimum distance $d\geq 6$ by employing intersection subspaces deduced from the direct sum. This method will sufficiently increase the number of possible repair groups of dimensional optimal LRCs, and thus efficiently expanding the range of the construction parameters while keeping the largest code rates compared with all known binary linear LRCs with minimum distance $d\geq 6$ and locality $2^b$ ($b\geq 3$).