Geometric construction of k-optimal locally repairable codes
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.