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. Also, it is 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 this also gives a constructive method to find locally repairable codes. Constructions are given of three infinite classes of optimal vector-linear locally repairable codes over an alphabet of small size, not depending on the size of the code. 
Toni Ernvall, Thomas Westerb\"ack, Camilla Hollanti and Ragnar Freij
cs.IT math.IT