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The locally repairable codes (LRCs) were introduced to correct erasures efficiently in distributed storage systems. LRCs are extensively studied recently. In this paper, we first deal with the open case remained in q and derive an improved upper bound for the minimum distances of LRCs. We also give an explicit construction for LRCs attaining this bound. Secondly, we consider the constructions of LRCs with any locality and availability which have high code rate and minimum distance as large asarXiv:1506.04822v1 fatcat:atvy4pxribd6bg3lizv7vaa5ju