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 first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon...

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Main Authors: Li, Xudong, Ma, Liming, Xing, Chaoping
Other Authors: School of Physical and Mathematical Sciences
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Language:English
Published: 2020
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Online Access:https://hdl.handle.net/10356/143232
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spelling sg-ntu-dr.10356-1432322023-02-28T19:48:40Z Optimal locally repairable codes via elliptic curves Li, Xudong Ma, Liming Xing, Chaoping School of Physical and Mathematical Sciences Science::Mathematics Automorphism Group J-invariants 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. Ministry of Education (MOE) Accepted version X. Li was supported in part by the Key Natural Science Fund of Sichuan Education Department under Grant 13ZA0031, in part by the Key Scientific Research Fund of Xihua University under Grant R1222628, in part by the Spring Plan of Ministry of Education under Grant Z2017065, and in part by the NSFC Project under Grant U1433130. L. Ma was supported by NSFC under Grant 11501493 and in part by the China Scholarship Council. C. Xing was supported by the Singapore MOE Tier 1 Research under Grant RG25/16. 2020-08-14T03:59:23Z 2020-08-14T03:59:23Z 2018 Journal Article Li, X., Ma, L., & Xing, C. (2019). Optimal locally repairable codes via elliptic curves. IEEE Transactions on Information Theory, 65(1), 108-117. doi:10.1109/TIT.2018.2844216 0018-9448 https://hdl.handle.net/10356/143232 10.1109/TIT.2018.2844216 2-s2.0-85048179197 1 65 108 117 en IEEE Transactions on Information Theory © 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at: https://doi.org/10.1109/TIT.2018.2844216. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics
Automorphism Group
J-invariants
spellingShingle Science::Mathematics
Automorphism Group
J-invariants
Li, Xudong
Ma, Liming
Xing, Chaoping
Optimal locally repairable codes via elliptic curves
description 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.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Li, Xudong
Ma, Liming
Xing, Chaoping
format Article
author Li, Xudong
Ma, Liming
Xing, Chaoping
author_sort Li, Xudong
title Optimal locally repairable codes via elliptic curves
title_short Optimal locally repairable codes via elliptic curves
title_full Optimal locally repairable codes via elliptic curves
title_fullStr Optimal locally repairable codes via elliptic curves
title_full_unstemmed Optimal locally repairable codes via elliptic curves
title_sort optimal locally repairable codes via elliptic curves
publishDate 2020
url https://hdl.handle.net/10356/143232
_version_ 1759857731216343040