A locally recoverable (LRC) code is a code over a finite field ${\mathbb{F}}_q$ such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more than one erasure, which are subfield-subcodes of some J-affine variety codes. For these LRC codes, we compute localities $(r,\delta )$ that determine the minimum size of a set $\bar{R}$ of positions so that any $\delta -1$ erasures in $\bar{R}$ can be recovered from the remaining r coordinates in this set. We also show that some of these LRC codes with lengths $n\gg q$ are $(\delta -1)$-optimal.

中文翻译：

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本地可恢复的J-仿射变种代码

本地可恢复（LRC）代码是有限域上的代码 ${\mathbb{F}}_q$这样就可以从该代码字中的少量其他坐标中恢复该代码字的任何擦除坐标。我们构造了校正多个擦除的LRC码，它们是一些J仿射变种码的子字段子码。对于这些LRC代码，我们计算位置$（\mathrm{[R}，\delta ）$ 确定一组的最小大小 $\stackrel{〜}{\mathrm{[R}}$ 的位置，以便任何 $\delta -\mathrm{1个}$ 删除 $\stackrel{〜}{\mathrm{[R}}$可以从该集合中剩余的r坐标中恢复。我们还显示了其中一些LRC代码的长度$ñ\gg q$ 是 $（\delta -\mathrm{1个}）$-最佳。