Locally recoverable J-affine variety codes

Abstract

A locally recoverable (LRC) code is a code over a finite field Fq 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.