Locally recoverable $J$-affine variety codes

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.