Constructions and Properties of Linear Locally Repairable Codes

In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. It is also shown that, with high probability, a random matrix with a few extra columns guaranteeing the locality property is a generator matrix for a locally repairable code with a good minimum distance. The proof of the result provides a constructive method to find locally repairable codes. Finally, constructions of three infinite classes of optimal vector-linear locally repairable codes over a small alphabet independent of the code size are given.