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Constructions of optimal locally recoverable codes via Dickson polynomials

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Abstract

In 2014, Tamo and Barg have presented in a very remarkable paper a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality). The key ingredients for constructing such optimal linear LRC codes are the so-called r-good polynomials, where r is equal to the locality of the LRC code. In 2018, Liu et al. presented two general methods of designing r-good polynomials by using function composition, which led to three new constructions of r-good polynomials. Next, Micheli provided a Galois theoretical framework which allows to construct r-good polynomials. The well-known Dickson polynomials form an important class of polynomials which have been extensively investigated in recent years in different contexts. In this paper, we provide new methods of designing r-good polynomials based on Dickson polynomials. Such r-good polynomials provide new constructions of optimal LRC codes.

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Notes

  1. For other l’s, the set I is given in Remark 5.

  2. For other l’s, the set I is given in Remark 9.

References

  1. Agarwal A., Barg A., Hu S., Mazumdar A., Tamo I.: Combinatorial alphabet-dependent bounds for locally recoverable codes. IEEE Trans. Inf Theory 64(5), 3481–3492 (2018).

    Article  MathSciNet  Google Scholar 

  2. Cadambe V.R., Mazumdar A.: Bounds on the size of locally recoverable codes. IEEE Trans. Inf Theory 61(11), 5787–5794 (2015).

    Article  MathSciNet  Google Scholar 

  3. Chou W.S., Gomez-Calderon J., Mullen G.L.: Value sets of Dickson polynomials over finite fields. J. Numb. Theory 30(3), 334–344 (1988).

    Article  MathSciNet  Google Scholar 

  4. Gopalan P., Huang C., Simitci H., Yekhanin S.: On the locality of codeword symbols. IEEE Trans. Inf Theory 58(11), 6925–6934 (2012).

    Article  MathSciNet  Google Scholar 

  5. Jin L.: Explicit construction of optimal locally recoverable codes of distance 5 and 6 via binary constant weight codes. IEEE Trans. Inf Theory 65(8), 4658–4663 (2019).

    Article  MathSciNet  Google Scholar 

  6. Kruglik S., Nazirkhanova K., Frolov A.: New bounds and generalizations of locally recoverable codes with availability. IEEE Trans. Inf Theory 65(7), 4156–4166 (2019).

    Article  MathSciNet  Google Scholar 

  7. Lausch H., Nöbauer W.: Algebra of Polynomials. North-Holland, Amsterdam (1973).

    MATH  Google Scholar 

  8. Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, New York (1997).

    Google Scholar 

  9. Lidl R., Mullen G.L., Turnwald G.: Dickson Polynomials, Pitman Monographs in Pure and Applied Mathematics, vol. 65. Addison-Wesley, Reading (1993).

    MATH  Google Scholar 

  10. Liu J., Mesnager S., Chen L.: New constructions of optimal locally recoverable codes via good polynomials. IEEE Trans. Inf Theory 64(2), 889–899 (2018).

    Article  MathSciNet  Google Scholar 

  11. Micheli G.: Constructions of locally recoverable codes which are optimal. IEEE Trans. Inf Theory 66(1), 167–175 (2020).

    Article  MathSciNet  Google Scholar 

  12. Papailiopoulos D.S., Dimakis A.G.: Locally repairable codes. IEEE Trans. Inf Theory 60(10), 5843–5855 (2014).

    Article  MathSciNet  Google Scholar 

  13. Silberstein N., Zeh A.: Anticode-based locally repairable codes with high availability. Des. Codes Cryptogr. 86(2), 419–445 (2018).

    Article  MathSciNet  Google Scholar 

  14. Tamo I., Barg A.: A family of optimal locally recoverable codes. IEEE Trans. Inf Theory 60(8), 4661–4676 (2014).

    Article  MathSciNet  Google Scholar 

  15. Tamo I., Barg A., Frolov A.: Bounds on the parameters of locally recoverable codes. IEEE Trans. Inf Theory 62(6), 3070–3083 (2016).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grants 61902276 and 61872435). The authors would like to thank the anonymous reviewers for their valuable comments which have highly improved the manuscript. The first and the second authors would also like to thank Gaojun Luo for the discussion on r-good polynomials via Dickson polynomials in Hangzhou, China.

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Authors and Affiliations

  1. School of Cybersecurity, College of Intelligence and Computing, Tianjin University, Tianjin, 300350, China

    Jian Liu

  2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China

    Jian Liu & Deng Tang

  3. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  4. LAGA UMR 7539, CNRS, Sorbonne Paris Cité, University of Paris XIII, 93430, Villetaneuse, France

    Sihem Mesnager

  5. Telecom ParisTech, 91120, Palaiseau, France

    Sihem Mesnager

  6. School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Deng Tang

Authors
  1. Jian Liu
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  2. Sihem Mesnager
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  3. Deng Tang
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Correspondence to Sihem Mesnager.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography 2019”.

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Liu, J., Mesnager, S. & Tang, D. Constructions of optimal locally recoverable codes via Dickson polynomials. Des. Codes Cryptogr. 88, 1759–1780 (2020). https://doi.org/10.1007/s10623-020-00731-0

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  • DOI: https://doi.org/10.1007/s10623-020-00731-0

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