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Like classical block codes, a locally repairable code also obeys theSingleton-type bound (we call a locally repairable code {\it optimal} if itachieves the Singleton-type bound). In the breakthrough work of \cite{TB14},several classes of optimal locally repairable codes were constructed viasubcodes 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 wasproved through extension of construction in \cite{TB14} that length of $q$-aryoptimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly,\cite{BHHMV16} presented a few examples of $q$-ary optimal locally repairablecodes of small distance and locality with code length achieving roughly $q^2$.Very recently, it was further shown in \cite{LMX17} that there exist $q$-aryoptimal locally repairable codes with length bigger than $q+1$ and distancepropositional to $n$.  Thus, it becomes an interesting and challenging problem to construct newfamilies 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 ofdistance $3$ and $4$ with unbounded length (i.e., length of the codes isindependent of the code alphabet size). Our technique is through cyclic codeswith particular generator and parity-check polynomials that are carefullychosen.

Optimal locally repairable codes of distance $3$ and $4$ via cyclic codes