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Optimal RS-like LRC codes of arbitrary length

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Abstract

RS-like locally recoverable (LRC) codes have construction based on the classical construction of Reed–Solomon (RS) codes, where codewords are obtained as evaluations of suitably chosen polynomials. These codes were introduced by Tamo and Barg (IEEE Trans Inf Theory 60(8):4661–4676, 2014) where they assumed that the length n of the code is divisible by \(r+1\), where r is the locality of the code. They also proposed a construction with this condition lifted to \(n \ne 1 \bmod (r+1)\). In a recent paper, Kolosov et al. (Optimal LRC codes for all lenghts \(n \le q\), arXiv:1802.00157, 2018) have given an explicit construction of optimal LRC codes with this lifted condition on n. In this paper we remove any such restriction on n completely, i.e., we propose constructions for q-ary RS-like LRC codes of any length \(n \le q\). Further, we show that the codes constructed by the proposed construction are optimal LRC codes for their parameters.

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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., Mazumdar, A.: An upper bound on the size of locally recoverable codes. In: 2013 International Symposium on Network Coding (NetCod), pp. 1–5. IEEE, New York (2013)

  3. 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 

  4. Goparaju, S., Calderbank, R.: Binary cyclic codes that are locally repairable. In: 2014 IEEE International Symposium on Information Theory, pp. 676–680. IEEE, New York (2014)

  5. Guruswami, V., Xing, C., Yuan, C.: How long can optimal locally repairable codes be? IEEE Trans. Inf. Theory 65(6), 3662–3670 (2019)

    Article  MathSciNet  Google Scholar 

  6. 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 

  7. Kolosov, O., Barg, A., Tamo, I., Yadgar, G.: Optimal LRC codes for all lenghts \(n \le q\). arXiv preprint arXiv:1802.00157 (2018)

  8. Liu, J., Mesnager, S., Chen, L.: New constructions of optimal locally recoverable codes via good polynomials. IEEE Trans. Inf. Theory 64(2), 889–899 (2017)

    Article  MathSciNet  Google Scholar 

  9. Liu, J., Mesnager, S., Tang, D.: Constructions of optimal locally recoverable codes via Dickson polynomials. In: Designs, Codes and Cryptography, pp. 1–22 (2020)

  10. Micheli, G.: Constructions of locally recoverable codes which are optimal. IEEE Trans. Inf. Theory 66(1), 167–175 (2019)

    Article  MathSciNet  Google Scholar 

  11. Pamies-Juarez, L., Hollmann, H.D., Oggier, F.: Locally repairable codes with multiple repair alternatives. In: 2013 IEEE International Symposium on Information Theory, pp. 892–896. IEEE, New York (2013)

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

    Article  MathSciNet  Google Scholar 

  13. Prakash, N., Lalitha, V., Kumar, P.V.: Codes with locality for two erasures. In: 2014 IEEE International Symposium on Information Theory, pp. 1962–1966. IEEE, New York (2014)

  14. Rawat, A.S., Papailiopoulos, D.S., Dimakis, A.G., Vishwanath, S.: Locality and availability in distributed storage, pp. 4481–4493. IEEE, New York (2016)

  15. Silberstein, N., Rawat, A.S., Koyluoglu, O.O., Vishwanath, S.: Optimal locally repairable codes via rank-metric codes. In: 2013 IEEE International Symposium on Information Theory, pp. 1819–1823. IEEE, New York (2013)

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

    Article  MathSciNet  Google Scholar 

  17. 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 

  18. Tamo, I., Barg, A., Goparaju, S., Calderbank, R.: Cyclic LRC codes, binary LRC codes, and upper bounds on the distance of cyclic codes. Int. J. Inf. Coding Theory 3 (2016). https://doi.org/10.1504/IJICOT.2016.079496

  19. Tamo, I., Papailiopoulos, D.S., Dimakis, A.G.: Optimal locally repairable codes and connections to matroid theory. IEEE Trans. Inf. Theory 62(12), 6661–6671 (2016)

    Article  MathSciNet  Google Scholar 

  20. Tan, P., Zhou, Z., Yan, H., Parampalli, U.: Optimal cyclic locally repairable codes via cyclotomic polynomials. IEEE Commun. Lett. 23(2), 202–205 (2018)

    Article  Google Scholar 

  21. Xing, C., Yuan, C.: Construction of optimal locally recoverable codes and connection with hypergraph. arXiv preprint arXiv:1811.09142 (2018)

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Acknowledgements

The authors would like to thank the anonymous referees for their careful reading of the manuscript and their many valuable comments and suggestions, which greatly improved the manuscript. The first author would like to thank University Grant Commission, India, for providing financial support.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

    Charul Rajput & Maheshanand Bhaintwal

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  1. Charul Rajput
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  2. Maheshanand Bhaintwal
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Correspondence to Maheshanand Bhaintwal.

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Rajput, C., Bhaintwal, M. Optimal RS-like LRC codes of arbitrary length. AAECC 31, 271–289 (2020). https://doi.org/10.1007/s00200-020-00430-2

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  • DOI: https://doi.org/10.1007/s00200-020-00430-2

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