Anticode-based locally repairable codes with high availability

This paper presents constructions of new families of locally repairable codes (LRCs) with small locality and high availability, where each code symbol can be recovered by using many (exponential in the dimension of the code) disjoint small sets (of size 2 or 3) of other code symbols. Following the method of Farrell, the generator matrices of our LRCs are obtained by deleting certain columns from the generator matrix of the Simplex code, where the deleted columns form different anticodes. Most of the resulting codes, defined over any finite field and in particular over the binary field, are optimal either with respect to the Griesmer bound, or with respect to the Cadambe–Mazumdar bound for LRCs, or both.