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.

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