  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 $q$.
