Skip to main content
SpringerLink
Log in

Some New Constructions of Difference Systems of Sets

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein, which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we describe four direct constructions of optimal DSSs from finite projective geometries and present a recursive construction of DSSs by extending the known construction. As a consequence, new infinite families of optimal DSSs can be obtained.

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 Availability

This manuscript has no associated data.

References

  1. Arasu, K.T., Ding, C., Helleseth, T., Kumar, D.V., Martinsen, H.M.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inform. Theory 47(7), 2934–2943 (2001)

    Article  MathSciNet  Google Scholar 

  2. Chang, Y., Ding, C.: Constructions of external difference families and disjoint difference families. Des. Codes Cryptogr. 40(2), 167–185 (2006)

    Article  MathSciNet  Google Scholar 

  3. Chisaki, S., Kimura, Y., Miyamoto, N.: A recursive construction for difference systems of sets. Des. Codes Cryptogr. 87(5), 1059–1068 (2019)

    Article  MathSciNet  Google Scholar 

  4. Chisaki, S., Miyamoto, N.: Difference systems of sets and a collection of \(3\)-subsets in a finite field of order \(p\). Finite Fields Appl. 34, 75–94 (2015)

    Article  MathSciNet  Google Scholar 

  5. Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Combin. Theory Ser. A 113(7), 1526–1535 (2006)

    Article  MathSciNet  Google Scholar 

  6. Fan, C., Lei, J., Chang, Y.: Constructions of difference systems of sets and disjoint difference families. IEEE Trans. Inform. Theory 54(7), 3195–3201 (2008)

    Article  MathSciNet  Google Scholar 

  7. Fan, C., Lei, J.: Constructions of difference systems of sets from finite projective geometry. IEEE Trans. Inform. Theory 58(1), 130–138 (2012)

    Article  MathSciNet  Google Scholar 

  8. Fuji-Hara, R., Munemasa, A., Tonchev, V.D.: Hyperplane partitions and difference systems of sets. J. Combin. Theory Ser. A 113(8), 1689–1698 (2006)

    Article  MathSciNet  Google Scholar 

  9. Fujiwara, Y., Fuji-Hara, R.: Frequency hopping sequences with optimal auto-and cross-correlation properties and related codes, In: Proceedings of Tenth International Workshop Algebraic Combinatorial Coding Theory, 83-96 (2006)

  10. Jungnickel, D., Pott, A., Smith, K.W.: Difference sets. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, pp. 419–436. CRC Press, Boca Raton, FL (2007)

    Google Scholar 

  11. Lei, J., Fan, C.: Optimal difference systems of sets and partition-type cyclic difference packings. Des. Codes Cryptogr. 58(2), 135–153 (2011)

    Article  MathSciNet  Google Scholar 

  12. Lempel, A., Cohn, M., Eastman, W.L.: A class of binary sequences with optimal autocorrelation properties. IEEE Trans. Inform. Theory 23(1), 38–42 (1977)

    Article  MathSciNet  Google Scholar 

  13. Levenshtein, V.I.: One method of constructing quasilinear codes providing synchronization in the presence of errors. Prob. Inform. Transm. 7(3), 215–222 (1971)

    Google Scholar 

  14. Mutoh, Y., Tonchev, V.: Difference systems of sets and cyclotomy. Discret. Math. 308(4), 2959–2969 (2008)

    Article  MathSciNet  Google Scholar 

  15. Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933)

    Article  Google Scholar 

  16. Qi, M., Xiong, S., Yuan, J., Rao, W., Zhong, L.: On some new difference systems of sets constructed from the cyclotomic classes of order 12. Discret. Math. 340(10), 2531–2537 (2017)

    Article  MathSciNet  Google Scholar 

  17. Storer, T.: Cyclotomy and Difference Sets. Lectures in Advanced Mathematics. Markham Publishing Company, Chicago (1967)

  18. Tonchev, V.: Difference systems of sets and code synchronization. Rend. Sem. Mat. Messina Ser. II 9, 217–226 (2003)

  19. Tonchev, V.: Partitions of difference sets and code synchronization. Finite Fields Appl. 11(3), 601–621 (2005)

    Article  MathSciNet  Google Scholar 

  20. Wang, H.: A new bound for difference systems of sets. J. Comb. Math. Comb. Comput. 58, 161–167 (2006)

    MathSciNet  Google Scholar 

  21. Wang, X., Wang, J.: Optimal difference systems of sets and difference sets. Aequ. Math. 82(1–2), 155–164 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous reviewers for many valuable suggestions that allowed us to improve the paper.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China

    Shuyu Shen & Jingjun Bao

Authors
  1. Shuyu Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingjun Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jingjun Bao.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

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

Shen, S., Bao, J. Some New Constructions of Difference Systems of Sets. Graphs and Combinatorics 40, 6 (2024). https://doi.org/10.1007/s00373-023-02729-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-023-02729-6

Keywords

Search

Navigation