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Repairing multiple failures for algebraic geometry codes

  • School of Mathematics and Physics,Anqing Normal University,Anqing 246133,China

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https://doi.org/10.3969/j.issn.0253-2778.2020.02.009
  • Received Date: 17 September 2018
  • Accepted Date: 10 April 2019
  • Rev Recd Date: 10 April 2019
  • Publish Date: 28 February 2020
    Abstract

  • Abstract

    Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable codes.Thus,the number of nodes is upper-bounded by 2b,where b is the bits of data stored in each node.From both theoretical and practical points of view,it is natural to consider regenerating codes that nearly have minimum storage of data,and meanwhile,the number of nodes is unbounded.Aiming at the problem,Jin et al.constructed the regenerating codes by algebraic geometry codes,which generalized the repairing algorithm of Reed-Solomon codes by Guruswami and Wotters.This paper mainly gives a construction to repair multiple failures for algebraic geometry codes,which extends the framework of repairing one failure for the regenerating codes. The results generalize some quite recent results in which regenerating codes,for instance,Reed-Solomon codes and scalar codes with multiple erasures.

    Abstract

    Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable codes.Thus,the number of nodes is upper-bounded by 2b,where b is the bits of data stored in each node.From both theoretical and practical points of view,it is natural to consider regenerating codes that nearly have minimum storage of data,and meanwhile,the number of nodes is unbounded.Aiming at the problem,Jin et al.constructed the regenerating codes by algebraic geometry codes,which generalized the repairing algorithm of Reed-Solomon codes by Guruswami and Wotters.This paper mainly gives a construction to repair multiple failures for algebraic geometry codes,which extends the framework of repairing one failure for the regenerating codes. The results generalize some quite recent results in which regenerating codes,for instance,Reed-Solomon codes and scalar codes with multiple erasures.
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    • References(12)
  • [1]
    DIMAKIS A G,GODFREY P B,WU Y,et al.Network coding for distributed storage systems[J].IEEE Trans Inf Theory,2010,56(9): 4539-4551.
    [2]
    DIMAKIS A G,RAMCHANDRAN K,WU Y,et al.A survey on network codes for distributed storage[J].Proc IEEE,2011,99(3): 476-489.
    [3]
    GURUSWAMI V,WOOTTERS M.Repairing Reed-Solomon codes[J].IEEE Trans Inf Theory,2016,63(9): 5684-5698.
    [4]
    DAU H,DUURSMA I,KIAH H M,et al.Repairing Reed-Solomon codes with multiple erasures[DB/OL].[2018-09-01]. https://arxiv.org/abs/1612.01361.
    [5]
    DAU H,DUURSMA I,KIAH H M,et al.Repairing Reed-Solomon codes with two erasures[DB/OL].[2018-09-01].https://arxiv.org/abs/1701.07118.
    [6]
    CADAMBE V R,HUANG C,LI J.Permutation code: Optimal exact-repair of a single failed node in MDS code based distributed storage systems[C]// 2011 IEEE International Symposium on Information Theory Proceedings.IEEE,2011: 1225-1229.
    [7]
    GOPPAV D.Codes on algebraic curves[J].Dokl Akad Nauk SSSR,1981,24(1): 170-172.(in Russian)
    [8]
    STICHTENOTH H.Algebraic Function Fields and Codes[M].Berlin: Springer,1993.
    [9]
    LI J,TANG X,TIAN C.Enabling all-node-repair in minimum storage regenerating codes[DB/OL].[2018-09-01].https://arxiv.org/abs/1604.07671.
    [10]
    JIN L,LUO Y,XING C.Repairing algebraic geometry codes[J].IEEE Trans Inf Theory,2018,64(2): 900-908.
    [11]
    LIDL R,NIEDERREITER H.Finite Fields[M].Cambridge: Cambridge University Press,2003.
    [12]
    MARDIA J,BARTANY B,WOOTTERS M.Repairing multiple failures for scalar MDS codes[DB/OL].[2018-09-01]. https://arxiv.org/abs/1707.02241.)
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    [1]
    DIMAKIS A G,GODFREY P B,WU Y,et al.Network coding for distributed storage systems[J].IEEE Trans Inf Theory,2010,56(9): 4539-4551.
    [2]
    DIMAKIS A G,RAMCHANDRAN K,WU Y,et al.A survey on network codes for distributed storage[J].Proc IEEE,2011,99(3): 476-489.
    [3]
    GURUSWAMI V,WOOTTERS M.Repairing Reed-Solomon codes[J].IEEE Trans Inf Theory,2016,63(9): 5684-5698.
    [4]
    DAU H,DUURSMA I,KIAH H M,et al.Repairing Reed-Solomon codes with multiple erasures[DB/OL].[2018-09-01]. https://arxiv.org/abs/1612.01361.
    [5]
    DAU H,DUURSMA I,KIAH H M,et al.Repairing Reed-Solomon codes with two erasures[DB/OL].[2018-09-01].https://arxiv.org/abs/1701.07118.
    [6]
    CADAMBE V R,HUANG C,LI J.Permutation code: Optimal exact-repair of a single failed node in MDS code based distributed storage systems[C]// 2011 IEEE International Symposium on Information Theory Proceedings.IEEE,2011: 1225-1229.
    [7]
    GOPPAV D.Codes on algebraic curves[J].Dokl Akad Nauk SSSR,1981,24(1): 170-172.(in Russian)
    [8]
    STICHTENOTH H.Algebraic Function Fields and Codes[M].Berlin: Springer,1993.
    [9]
    LI J,TANG X,TIAN C.Enabling all-node-repair in minimum storage regenerating codes[DB/OL].[2018-09-01].https://arxiv.org/abs/1604.07671.
    [10]
    JIN L,LUO Y,XING C.Repairing algebraic geometry codes[J].IEEE Trans Inf Theory,2018,64(2): 900-908.
    [11]
    LIDL R,NIEDERREITER H.Finite Fields[M].Cambridge: Cambridge University Press,2003.
    [12]
    MARDIA J,BARTANY B,WOOTTERS M.Repairing multiple failures for scalar MDS codes[DB/OL].[2018-09-01]. https://arxiv.org/abs/1707.02241.)
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