Optimal Locally Repairable Codes Via Elliptic Curves

• Published in: IEEE transactions on information theory Vol. 65; no. 1; pp. 108 - 117
• Main Authors: Li, Xudong, Ma, Liming, Xing, Chaoping
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">j -invariants; Automorphism group; Codes; Elliptic curves; Elliptic functions; function fields; Linear codes; maximal curves; rational points; Reed-Solomon codes; Riemann-Roch spaces
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2018.2844216

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 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 by Tamo and Barg have length upper bounded by q. Recently, it was shown through extension of construction by Tamo and Barg that length of q-ary optimal locally repairable codes can be q+1 by Jin et al.. Surprisingly it was shown by Barg et al. 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 this 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.