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Constructions of optimal locally recoverable codes via Dickson polynomials

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. 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.

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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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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. (2020). https://doi.org/10.1007/s10623-020-00731-0

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Keywords

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

Mathematics Subject Classification

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