Skip to main content
SpringerLink
Log in

The minimum locality of linear codes

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

Abstract

Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum locality of linear codes. In addition, the minimum locality of many known families of linear codes has not been studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum locality of linear codes, and investigates the minimum locality of a number of families of linear codes, such as q-ary Hamming codes, q-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. To this end, the concepts of linear locality and minimum linear locality are specified. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper.

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

References

  1. Abatangelo V., Larato B.: Near-MDS codes arising from algebraic curves. Discret. Math. 301, 5–19 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  2. Abatangelo V., Larato B.: Elliptic near-MDS codes over \(\mathbf{F} _5\). Des. Codes Cryptgr. 46, 167–174 (2008).

    Article  MATH  Google Scholar 

  3. Assmus Jr., Mattson Jr.: New 5-designs. J. Comb. Theory, 6(2), 122–151 (1969)

  4. Assmus E.F. Jr., Key J.D.: Polynomial codes and finite geometries. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 1269–1343. Elsevier, Amsterdam (1998).

    Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, H., Fan, C., Miao, Y., Schwartz, M., Tang, X.: Optimal locally repairable codes: an improved bound and constructions. arXiv:2011.04966v1 [cs.IT]

  8. 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 

  9. Chen B., Chen J.: A construction of optimal \((r, \delta )\)-locally recoverable codes. IEEE Access 7, 180349–180353 (2019).

    Article  Google Scholar 

  10. De Boer M.A.: Almost MDS codes. Des. Codes Cryptogr. 9, 143–155 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding C.: Codes from Difference Sets. World Scientific, Singapore (2015).

    Google Scholar 

  12. Ding C., Heng Z.: The subfield codes of ovoid codes. IEEE Trans. Inf. Theory 65(8), 4715–4729 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  13. Ding C., Tang C.: Designs from Linear Codes, 2nd edn World Scientific, Singapore (2018).

    Book  MATH  Google Scholar 

  14. Ding C., Tang C.: Infinite families of near MDS codes holding \(t\)-designs. IEEE Trans. Inf. Theory 66(9), 5419–5428 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  15. Dodunekov S., Landgev I.: On near-MDS codes. J. Geom. 54(1–2), 30–43 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  16. Dodunekov S.M., Landjev I.N.: Near-MDS codes over some small fields. Discret. Math. 213(1–3), 55–65 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  17. Faldum A., Willems W.: Codes of small defect. Des. Codes Cryptogr. 10, 341–350 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  18. Giulietti M.: On the extendibility of near-MDS elliptic codes. AAECC 15, 1–11 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  19. 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  MATH  Google Scholar 

  20. Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de, 2007. Accessed 07 Jan 2021

  21. Grezet M., Hollanti C.: The complete hierarchical locality of the punctured Simplex code. Des. Codes Cryptogr. 89, 1–21 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo, A., Kopparty, S., Sudan, M.M.: New affine-invariant codes from lifting. In: Proceedings of the 4th Conference on Innovations in Theoretical Computer Science, pp. 529–540 (2013)

  23. Huang P., Yaakobi E., Uchikawa H., Siegel P.H.: Binary linear locally repairable codes. IEEE Trans. Inf. Theory 62(11), 6268–6283 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  24. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  MATH  Google Scholar 

  25. Jin L., Kan H.: Self-dual near MDS codes from elliptic curves. IEEE Trans. Inf. Theory 65(4), 2166–2170 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  26. Jin L., Kan H., Zhang Y.: Constructions of locally repairable codes with multiple recovering sets via rational function fields. IEEE Trans. Inf. Theory 66(1), 202–209 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu J., Mesnager S., Tang D.: Constructions of optimal locally recoverable codes via Dickson polynomials. Des. Codes Cryptogr. 88(2), 1759–1780 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu Y., Ding C., Tang C.: Shortened linear codes over finite fields. IEEE Trans. Inf. Theory 67(8), 5119–5132 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  29. 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  MATH  Google Scholar 

  30. Marcugini S., Milani A., Pambianco F.: NMDS codes of maximum length over \(\mathbf{F} _q\), \(8\le q \le 11,\). IEEE Trans. Inf. Theory 48(4), 963–966 (2002).

    Article  MATH  Google Scholar 

  31. Maschietti A.: Difference set and hyperovals. Des. Codes Cryptogr. 14, 89–98 (1998).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Tan P., Zhou Z., Sidorenko V., Parampalli U.: Two classes of optimal LRCs with information \((r, t)\)-locality. Des. Codes Cryptogr. 88(2), 1741–1757 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  35. Tang C., Ding C.: An infinite family of linear codes supporting \(4\)-designs. IEEE Trans. Inf. Theory 67(1), 244–254 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  36. Tang C., Ding C., Xiong M.: Codes, differentially \(\delta \)-uniform functions and \(t\)-designs. IEEE Trans. Inf. Theory 66(6), 3691–3703 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  37. Tong H., Ding Y.: Quasi-cyclic NMDS codes. Finite Fields Appl. 24, 45–54 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  38. Tsfasmann M.A., Vladut S.G.: Algebraic-Geometry Codes. Kluwer, Dordrecht (1991).

    Book  Google Scholar 

  39. Wang Q., Heng Z.: Near MDS codes from overall polynomials. Discret. Math. 344, 4 (2021).

    Article  Google Scholar 

  40. Wang A., Zhang Z., Liu M.: Achieving arbitrary locality and availability in binary codes. Proc. ISIT 2016, 1866–1870 (2015).

    Google Scholar 

  41. Xing, C., Yuan, C.: Construction of optimal locally recoverable codes and connection with hypergraph. Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), pp. 1–98 (2019)

Download references

Acknowledgements

The authors are very grateful to the reviewers and the Editor for their very detailed comments and suggestions that much improved the presentation and quality of this paper. The research of C. Fan and Z. Zhou was as supported by The National Natural Science Foundation of China (Grant Nos. 11971395, 62071397, and No. 62131016), and also by the Central Government Funds for Guiding Local Scientific and Technological Development under Grant 2021ZYD0001. The research of C. Ding was supported by the Research Grants Council of Hong Kong, under Grant No. 16301020. The research of C. Tang was supported by The National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds).

Author information

Authors and Affiliations

  1. School of Information Science and Technology, Yunnan Normal University, Kunming, 650500, China

    Pan Tan

  2. School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, China

    Cuiling Fan & Zhengchun Zhou

  3. Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    Cunsheng Ding

  4. School of Mathematics and Information, China West Normal University, Sichuan, Nanchong, 637002, China

    Chunming Tang

Authors
  1. Pan Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cuiling Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cunsheng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chunming Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhengchun Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cuiling Fan.

Additional information

Communicated by D. Panario.

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 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

Tan, P., Fan, C., Ding, C. et al. The minimum locality of linear codes. Des. Codes Cryptogr. 91, 83–114 (2023). https://doi.org/10.1007/s10623-022-01099-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-022-01099-z

Keywords

Mathematics Subject Classification

Search

Navigation