Skip to main content
Springer Nature Link
Log in

Two classes of optimal LRCs with information (r, t)-locality

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Locally repairable codes (LRCs) with (r, t)-locality have received considerable attention in recent years, since they are able to solve common problems in distributed storage systems such as repairing multiple node failures and managing hot data. Constructing LRCs with excellent parameters becomes an interesting research subject in distributed storage systems and coding theory. In this paper, we present two constructions of LRCs with information (r, t)-locality based on linear algebra and partial geometry, respectively. Both constructions generate LRCs with new parameters which are optimal with respect to the bound proposed by Rawat et al. (IEEE Trans Inf Theory 62(8):4481–4493, 2016).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Asi H., Yaakobi E.: Nearly optimal constructions of PIR and batch codes. IEEE Inter. Symp. Inf. Theory (ISIT) 65, 151–155 (2017).

    MATH  Google Scholar 

  2. Berman S.D.: Semisimple cyclic and abelian codes. Cybernetics 3, 21–30 (1967). (Russian).

    MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Cai H., Cheng M., Fan C., Tang X.: Optimal locally repairable systematic codes based on packings. IEEE Trans. Commun. 67(1), 39–49 (2019).

    Article  Google Scholar 

  5. Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs. CRC Press, Boca Raton (1996).

    Book  Google Scholar 

  6. Fazeli A., Vardy A., Yaakobi E.: PIR with low storage overhead: coding instead of replication. CoRR arXiv:1505.06241 (2015)

  7. 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 

  8. Hao J., Xia S.: Constructions of optimal binary locally repairable codes with multiple repair groups. IEEE Commun. Lett. 20(6), 1060–1063 (2016).

    Article  Google Scholar 

  9. Huang C., Chen M., Li J.: Pyramid codes: flexible schemes to trade space for access efficiency in reliable data storage systems. In: Proceedings of the IEEE International Symposium Network Computing and Applications (NCA 2007), pp. 79–86 (2007).

  10. Huang C., Simitci H., Xu Y., Ogus A., Calder B., Gopalan P., Li J., Yekhanin S., et al.: Erasure coding in windows azure storage. In: Usenix Annual Technical Conference, pp. 15–26. Boston, MA (2012)

  11. Ishai Y., Kushilevitz E., Ostrovsky R., Sahai A.: Batch codes and their applications. In: Proceedings of Annual ACM Symposium Theory of Computing, Chicago, IL, USA, pp. 262–271 (2004)

  12. Kamath G.M., Prakash N., Lalitha V., Kumar P.V.: Codes with local regeneration and erasure correction. IEEE Trans. Inf. Theory 60(8), 4637–4660 (2014).

    Article  MathSciNet  Google Scholar 

  13. Kruglik S., Frolov A.: Bounds and constructions of codes with all-symbol locality and availability. In: 2017 IEEE International Symposium on Information Theory (ISIT), pp. 1023–1027. IEEE (2017).

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

    Article  MathSciNet  Google Scholar 

  15. Prakash N., Kamath G.M., Lalitha V., Kumar P.V.: Optimal linear codes with a local-error-correction property. In: 2012 IEEE International Symposium on Information Theory Proceedings, pp. 2776–2780. IEEE (2012)

  16. Rawat A.S., Koyluoglu O.O., Silberstein N., Vishwanath S.: Optimal locally repairable and secure codes for distributed storage systems. IEEE Trans. Inf. Theory 60(1), 212–236 (2014).

    Article  Google Scholar 

  17. Rawat A.S., Papailiopoulos D.S., Dimakis A.G., Vishwanath S.: Locality and availability in distributed storage. IEEE Trans. Inf. Theory 62(8), 4481–4493 (2016).

    Article  MathSciNet  Google Scholar 

  18. Sathiamoorthy M., Asteris M., Papailiopoulos D., Dimakis A.G., Vadali R., Chen S., Borthakur D.: Xoring elephants: novel erasure codes for big data. Proc. VLDB Endow. 6, 325–336 (2013).

    Article  Google Scholar 

  19. Sidorenko V.R.: Class of correcting codes for errors with a lattice configuration. Probl. Inform. Transm. 12(3), 165–171 (1976).

    Google Scholar 

  20. Silberstein N., Etzion T., Schwartz M.: Locality and availability of array codes constructed from subspaces. IEEE Trans. Inf. Theory 65(5), 2648–2660 (2019).

    Article  MathSciNet  Google Scholar 

  21. Su Y.: On constructions of a class of binary locally repairable codes with multiple repair groups. IEEE Access 5, 3524–3528 (2017).

    Article  Google Scholar 

  22. Wang A., Zhang Z.: Repair locality with multiple erasure tolerance. IEEE Trans. Inf. Theory 60(11), 6979–6987 (2014).

    Article  MathSciNet  Google Scholar 

  23. Wang A., Zhang Z., Liu M.: Achieving arbitrary locality and availability in binary codes. In: 2015 IEEE International Symposium on Information Theory (ISIT), pp. 1866–1870. IEEE (2015)

Download references

Acknowledgements

The authors are very grateful to the co-chairs of WCC, Anne Canteaut and Felix Ulmer, and to the anonymous reviewers for their comments and suggestions. Special thanks to Alex Pott for bringing our attention to connection of proposed codes with geometries, we addressed these in Remarks. The authors also would like to thank the reviewers for bringing our attention to the connection of our codes with PIR/batch codes.

Author information

Authors and Affiliations

  1. School of Information Science and Technology, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

    Pan Tan

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

    Pan Tan

  3. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

    Zhengchun Zhou

  4. Institute for Communications Engineering, Technical University of Munich, Munich, Germany

    Vladimir Sidorenko

  5. School of Computing and Information Systems, The University of Melbourne, Melbourne, VIC, 3053, Australia

    Udaya Parampalli

Authors
  1. Pan Tan
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Zhengchun Zhou
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Vladimir Sidorenko
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Udaya Parampalli
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Zhengchun Zhou.

Additional information

Publisher's Note

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

The work of P. Tan and Z. Zhou was supported in part by the National Natural Science Foundation of China under Grants 61672028 and 11971395, and in part by Doctoral Innovation Fund Program of Southwest Jiaotong University.

V. Sidorenko is on leave from Institute for Information Transmission Problems, Russian Academy of Sciences. His work is supported by the Russian Government (Contract No 14.W03.31.0019).

U. Parampalli is supported in part by the Australian Research Councils Discovery under Grant DP150104473 and in part by the University of Melbourne’s 2014 International Research and Research Training Fund.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tan, P., Zhou, Z., Sidorenko, V. et al. Two classes of optimal LRCs with information (r, t)-locality. Des. Codes Cryptogr. 88, 1741–1757 (2020). https://doi.org/10.1007/s10623-020-00728-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-020-00728-9

Keywords

Mathematics Subject Classification