Optimal Locally Repairable Codes of Distance 3 and 4 via Cyclic Codes

Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code <italic>optimal</italic> if it achieves the Singleton-type bound). In the breakthrough work of Tamo and Barg, several classes of optimal locally repairable codes were constructed via subcodes of Reed–Solomon codes. Thus, the lengths of the codes given by Tamo and Barg are upper bounded by the code alphabet size <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>. Recently, it was proved through the extension of construction by Tamo and Barg that the length of <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary optimal locally repairable codes can be <inline-formula> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula> by Jin <italic>et al</italic>. Surprisingly, Barg <italic>et al.</italic> presented a few examples of <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary optimal locally repairable codes of small distance and locality with code length achieving roughly <inline-formula> <tex-math notation="LaTeX">$q^{2}$ </tex-math></inline-formula>. Very recently, it was further shown in the work of Li <italic>et al.</italic> that there exist <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary optimal locally repairable codes with the length bigger than <inline-formula> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula> and the distance proportional to <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. Thus, it becomes an interesting and challenging problem to construct new families of <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary optimal locally repairable codes of length bigger than <inline-formula> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula>. In this paper, we construct a class of optimal locally repairable codes of distances 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.