Locally recoverable J-affine variety codes

Abstract A locally recoverable (LRC) code is a code over a finite field 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 , δ ) that determine the minimum size of a set R ‾ of positions so that any δ − 1 erasures in R ‾ can be recovered from the remaining r coordinates in this set. We also show that some of these LRC codes with lengths n ≫ q are ( δ − 1 ) -optimal.