Construction of asymptotically good locally repairable codes via automorphism groups of function fields

Locally repairable codes have been investigated extensively in recent years due to practical application in distributed storage as well as theoretical interest. However, not much work on asymptotical behavior of locally repairable codes has been done until now. In particular, there is a little result on constructive lower bound on asymptotical behavior of locally repairable codes. In this paper, we extend the construction given in \cite{BTV17} via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality. Furthermore, we show that the Gilbert-Varshamov type bound on locally repairable codes can be improved for all sufficiently large alphabet size $q$.