Optimal locally repairable codes via elliptic curves
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Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg TB14 first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, q-ary optimal locally repairable codes from subcodes of Reed-Solomon codes given in TB14 have length upper bounded by q. Recently, it was shown through extension of construction in TB14 that length of q-ary optimal locally repairable codes can be q+1 in JMX17. Surprisingly it was shown in BHHMV16 that, unlike classical MDS codes, q-ary optimal locally repairable codes could have length bigger than q+1. Thus, it becomes an interesting and challenging problem to construct q-ary optimal locally repairable codes of length bigger than q+1. In the present paper, we make use of rich algebraic structures of elliptic curves to construct a family of q-ary optimal locally repairable codes of length up to q+2√(q). It turns out that locality of our codes can be as big as 23 and distance can be linear in length.
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