Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Constructions of optimal locally recoverable codes via Dickson polynomials


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.

This is a preview of subscription content, log in to check access.


  1. 1.

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

  2. 2.

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


  1. 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).

  2. 2.

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

  3. 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).

  4. 4.

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

  5. 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).

  6. 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).

  7. 7.

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

  8. 8.

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

  9. 9.

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

  10. 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).

  11. 11.

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

  12. 12.

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

  13. 13.

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

  14. 14.

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

  15. 15.

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

Download references


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.

Author information

Correspondence to Sihem Mesnager.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Mesnager, S. & Tang, D. Constructions of optimal locally recoverable codes via Dickson polynomials. Des. Codes Cryptogr. (2020).

Download citation


  • Locally recoverable code
  • Dickson polynomial
  • Polynomial over a finite field
  • Linear code

Mathematics Subject Classification

  • Primary: 11C08
  • 12-00 Secondary: 94B05