Locally recoverable codes on graphs

The locally recoverable codes (LRC codes) are linear codes with an important for applications property that every symbol of a codeword can be recovered from a small set of other symbols. Codeword symbols can be interpreted as servers with information. It is natural to define a topology of these servers as a graph so that every server can be recovered using neighborhood servers in this graph. The paper provides LRC code constructions for some topologies and bounds on the rate of such codes.