Abstract
Locally recoverable codes (LRCs) have a great significance in distributed storage systems, and have received considerable attention in recent years. In particular, it is a challenging task to construct optimal $(r,\delta)$ -LRCs, meaning $(r,\delta)$ -LRCs whose minimum distances attain Singleton-type bound. In this letter, we investigate the construction of a family of optimal $(r,\delta)$ -LRCs via generalized Reed-Solomon codes (GRS codes). Our strategy is to equip parity-check matrices for optimal $(r,\delta)$ -LRCs with the Vandermonde structure. Furthermore, based on these new optimal $(r,\delta)$ -LRCs we present a family of optimal locally recoverable codes with hierarchical locality (H-LRCs). The parameters of our results are not covered in the literature.
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