Geometric construction of k-optimal locally repairable codes
This work is significant for researchers and engineers in distributed storage systems, as it provides new methods for constructing more efficient and robust locally repairable codes, which are crucial for enhancing data availability and reducing repair costs.
This paper focuses on constructing k-optimal locally repairable codes (LRCs) to improve repair efficiency in distributed storage systems. The authors achieve this by introducing the s-Pasch configuration and using point-line relationships in PG(2,q) to construct q-ary k-optimal LRCs with minimum distance 6 and general locality r.
A linear code is referred to as a locally repairable code (LRC) with locality r if any erased code symbol can be recovered by accessing at most r other code symbols. LRCs are highly desirable for distributed storage systems to enhance repair efficiency. In this paper, we investigate LRCs with disjoint repair sets via the parity-check matrix method. Firstly, we propose a novel concept of the s-Pasch configuration and present a geometric characterization for the existence of LRCs with minimum distance 5 and locality 3. Subsequently, we construct k-optimal LRCs by exploiting the point-line relationship in PG(2,q). Finally, a family of q-ary k-optimal LRCs with minimum distance 6 and general locality r is constructed using partial r-spreads.