  Like classical block codes, a locally repairable code also obeys the
Singleton-type bound (we call a locally repairable code {\it optimal} if it
achieves the Singleton-type bound). In the breakthrough work of \cite{TB14},
several classes of optimal locally repairable codes were constructed via
subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in
\cite{TB14} are upper bounded by the code alphabet size $q$. Recently, it was
proved through extension of construction in \cite{TB14} that length of $q$-ary
optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly,
\cite{BHHMV16} presented a few examples of $q$-ary optimal locally repairable
codes of small distance and locality with code length achieving roughly $q^2$.
Very recently, it was further shown in \cite{LMX17} that there exist $q$-ary
optimal locally repairable codes with length bigger than $q+1$ and distance
propositional to $n$.
  Thus, it becomes an interesting and challenging problem to construct new
families of $q$-ary optimal locally repairable codes of length bigger than
$q+1$.
  In this paper, we construct a class of optimal locally repairable codes of
distance $3$ and $4$ with unbounded length (i.e., length of the codes is
independent of the code alphabet size). Our technique is through cyclic codes
with particular generator and parity-check polynomials that are carefully
chosen.
