🤖 AI Summary
This work addresses the efficiency and optimality of locally repairable codes (LRCs) in distributed storage by systematically investigating LRCs with disjoint repair sets through a parity-check matrix approach. Leveraging projective geometry PG(2,q), partial r-spread structures, and a newly introduced s-Pasch configuration, the study provides the first geometric characterization for the existence of LRCs with locality three and minimum distance five. Furthermore, it constructs a family of q-ary k-optimal LRCs achieving minimum distance six for arbitrary locality r, attaining the theoretical bound on code length versus minimum distance and thereby realizing parameter optimality.
📝 Abstract
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.