  Recent years, several new types of codes were introduced to provide
fault-tolerance and guarantee system reliability in distributed storage
systems, among which locally repairable codes (LRCs for short) have played an
important role.
  A linear code is said to have locality $r$ if each of its code symbols can be
repaired by accessing at most $r$ other code symbols. For an LRC with length
$n$, dimension $k$ and locality $r$, its minimum distance $d$ was proved to
satisfy the Singleton-like bound $d\leq n-k-\lceil k/r\rceil+2$. Since then,
many works have been done for constructing LRCs meeting the Singleton-like
bound over small fields.
  In this paper, we study quaternary LRCs meeting Singleton-like bound through
a parity-check matrix approach. Using tools from finite geometry, we provide
some new necessary conditions for LRCs being optimal. From this, we prove that
there are $27$ different classes of parameters for optimal quaternary LRCs.
Moreover, for each class, explicit constructions of corresponding optimal
quaternary LRCs are presented.
