Advertisement

On Good Polynomials over Finite Fields for Optimal Locally Recoverable Codes

  • Sihem MesnagerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11445)

Abstract

[This is an extended abstract of the paper [3]] A locally recoverable (LRC) code is a code that enables a simple recovery of an erased symbol by accessing only a small number of other symbols. LRC codes currently form one of the rapidly developing topics in coding theory because of their applications in distributed and cloud storage systems. In 2014, Tamo and Barg have presented in a very remarkable paper a family of LRC codes that attain the maximum possible (minimum) distance (given code length, cardinality, and locality). The key ingredient for constructing such optimal linear LRC codes is the so-called r-good polynomials, where r is equal to the locality of the LRC code. In this extended abstract, we review and discuss good polynomials over finite fields for constructing optimal LRC codes.

Keywords

Finite fields Good polynomials Locally recoverable codes Coding theory Storage 

Notes

Acknowledgements

The author is very grateful to Jian Liu for her valuable help. She also thanks the co-chairs program (in particular Claude Carlet for his careful reading) and the organizers of the conference C2SI 2019 for their nice invitation.

References

  1. 1.
    Gopalan, P., Huang, C., Simitci, H., Yekhanin, S.: On the locality of codeword symbols. IEEE Trans. Inf. Theory 58(11), 6925–6934 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hou, X.-D.: Explicit evaluation of certain exponential sums of binary quadratic functions. Finite Fields Appl. 13(4), 843–868 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Liu, J., Mesnager, S., Chen, L.: New constructions of optimal locally recoverable codes via good polynomials. IEEE Trans. Inf. Theory 62(2), 889–899 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Liu, J., Mesnager, S., Tang, D.: Constructions of optimal locally recoverable codes via Dickson polynomials. In: Workshop WCC 2019: The Eleventh International Workshop on Coding and Cryptography (2019)Google Scholar
  5. 5.
    Micheli, G.: Constructions of locally recoverable codes which are optimal. arXiv:1806.11492 [cs.IT] (2018)
  6. 6.
    Lidl, R., Niederreiter, H.: Finite Fields, Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
  7. 7.
    Papailiopoulos, D.S., Dimakis, A.G.: Locally repairable codes. IEEE Trans. Inf. Theory 60(10), 5843–5855 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Tamo, I., Barg, A.: A family of optimal locally recoverable codes. IEEE Trans. Inf. Theory 60(8), 4661–4676 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.LAGA, Department of Mathematics, University of Paris VIII and Paris XIII, UMR 7539; CNRS and Telecom ParisTechSaint-Denis cedex 02France

Personalised recommendations