Skip to main content
SpringerLink
Log in

Optimal binary and ternary locally repairable codes with minimum distance 6

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

Abstract

A locally repairable code (LRC) is a code that can recover any symbol of a codeword by reading at most \(r \) other symbols, denoted by \(r \)-LRC. In this paper, we study binary and ternary linear LRCs with disjoint repair groups and minimum distance \(d \) = 6. Using the intersection subspaces technique, we explicitly construct dimensional optimal LRCs. First, based on the intersection subspaces constructed by \(t \)-spread, a construction of binary LRCs is designed. Particularly, a class of binary linear LRCs with \(r \) = 11 is optimal in terms of achieving a sphere-packing type upper bound. Next, by using the Kronecker product of two matrices, two classes of dimensional optimal ternary LRCs with small locality (\(r \) = 3, 5) are presented. Compared to previous results, our construction is more flexible regarding code parameters. Finally, we also discuss the parameters of a code obtained by applying a shortening operation to our LRCs. We show that these shortened LRCs are also \(k \)-optimal.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availibility

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Asteris M., Papailiopoulos D., Dimakis A.G., Vadali R., Chen S., Borthakur D.: Xoring elephants: novel erasure codes for big data. In: Proceedings of the VLDB Endowment 6(5) (2013).

  2. Bu T.: Partitions of a vector space. Discret. Math. 31(1), 79–83 (1980).

    Article  MathSciNet  Google Scholar 

  3. Cadambe V.R., Mazumdar A.: An upper bound on the size of locally recoverable codes. In: International Symposium on Network Coding, NetCod 2013, Calgary, AB, Canada, June 7–9, 2013, pp. 1–5 (2013).

  4. 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 

  5. El-Zanati S., Jordon H., Seelinger G., Sissokho P., Spence L.: The maximum size of a partial 3-spread in a finite vector space over gf (2). Des. Codes Cryptogr. 54(2), 101–107 (2010).

    Article  MathSciNet  Google Scholar 

  6. Fang W., Chen B., Xia S.-T., Fu F.-W., Chen X.: Perfect LRCs and k-optimal LRCs. Des. Codes Cryptogr., 1–24 (2022).

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

  9. 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 

  10. Hao J., Xia S., Chen B.: On optimal ternary locally repairable codes. In: 2017 IEEE International Symposium on Information Theory (ISIT), pp. 171–175 (2017).

  11. Huang C., Simitci H., Xu Y., Ogus A., Calder B., Gopalan P., Li J., Yekhanin S.: Erasure coding in windows azure storage. In: 2012 USENIX Annual Technical Conference (USENIX ATC 12), pp. 15–26 (2012).

  12. Jin L., Ma L., Xing C.: Construction of optimal locally repairable codes via automorphism groups of rational function fields. IEEE Trans. Inf. Theory 66(1), 210–221 (2019).

    Article  MathSciNet  Google Scholar 

  13. Kim C., No J.-S.: New constructions of binary and ternary locally repairable codes using cyclic codes. IEEE Commun. Lett. 22(2), 228–231 (2017).

    Article  Google Scholar 

  14. Kim C., No J.-S.: New constructions of binary LRCs with disjoint repair groups and locality 3 using existing LRCs. IEEE Commun. Lett. 23(3), 406–409 (2019).

    Article  Google Scholar 

  15. Li X., Ma L., Xing C.: Optimal locally repairable codes via elliptic curves. IEEE Trans. Inf. Theory 65(1), 108–117 (2018).

    Article  MathSciNet  Google Scholar 

  16. Luo G., Cao X.: Constructions of optimal binary locally recoverable codes via a general construction of linear codes. IEEE Trans. Commun. 69(8), 4987–4997 (2021).

    Article  Google Scholar 

  17. Luo Y., Xing C., Yuan C.: Optimal locally repairable codes of distance 3 and 4 via cyclic codes. IEEE Trans. Inf. Theory 65(2), 1048–1053 (2019).

    Article  MathSciNet  Google Scholar 

  18. Ma J., Ge G.: Optimal binary linear locally repairable codes with disjoint repair groups. SIAM J. Discret. Math. 33(4), 2509–2529 (2019).

    Article  MathSciNet  Google Scholar 

  19. Nam M., Song H.: Binary locally repairable codes with minimum distance at least six based on partial \( t \)-spreads. IEEE Commun. Lett. 21(8), 1683–1686 (2017).

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Silberstein N., Zeh A.: Optimal binary locally repairable codes via anticodes. In: 2015 IEEE International Symposium on Information Theory (ISIT), pp. 1247–1251 (2015).

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. 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 

  25. 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 

  26. Wang A., Zhang Z., Lin D.: Bounds for binary linear locally repairable codes via a sphere-packing approach. IEEE Trans. Inf. Theory 65(7), 4167–4179 (2019).

    Article  MathSciNet  Google Scholar 

  27. Zeh A., Yaakobi E.: Optimal linear and cyclic locally repairable codes over small fields. In: 2015 IEEE Information Theory Workshop (ITW), pp. 1–5 (2015).

Download references

Acknowledgements

The authors express their sincere gratitude to the editor and the anonymous reviewers for their efforts in reviewing this article and providing constructive comments. This work was supported in part by National Key R &D Program of China under Grant 2022YFA1005000, and Fundings of SJTU-Alibaba Joint Research Lab on Cooperative Intelligent Computing.

Author information

Authors and Affiliations

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

    Wenqin Zhang & Yuan Luo

  2. Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, V6T1Z4, Canada

    Lele Wang

Authors
  1. Wenqin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lele Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Luo.

Additional information

Communicated by P. Charpin.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, W., Luo, Y. & Wang, L. Optimal binary and ternary locally repairable codes with minimum distance 6. Des. Codes Cryptogr. (2023). https://doi.org/10.1007/s10623-023-01341-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10623-023-01341-2

Keywords

Mathematics Subject Classification

Search

Navigation