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.

中文翻译：

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最优(r, δ)局部可恢复码的新构造

局部可恢复码（LRCs）在分布式存储系统中具有重要意义，近年来受到了相当多的关注。尤其是构建最优 $(r,\delta)$ -LRCs，意思是 $(r,\delta)$ - LRC 的最小距离达到单例类型的界限。在这封信中，我们研究了最优族的构建 $(r,\delta)$ -LRCs 通过广义 Reed-Solomon 码（GRS 码）。我们的策略是配备奇偶校验矩阵以获得最佳 $(r,\delta)$ -具有范德蒙德结构的LRC。此外，基于这些新的最优 $(r,\delta)$ -LRCs 我们提出了一系列具有分层局部性的最优局部可恢复代码（H-LRCs）。我们的结果的参数没有包含在文献中。