Optimal locally repairable codes via elliptic curves

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 [14] 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 [14] have length upper bounded by q. Recently, it was shown through extension of construction in [14] that length of q-ary optimal locally repairable codes can be q + 1 in [7]. Surprisingly it was shown in [2] 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.