Skip to main content
SpringerLink
Log in

New constructions of signed difference sets

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

Abstract

Signed difference sets have interesting applications in communications and coding theory. A \((v,k,\lambda )\)-difference set in a finite group G of order v is a subset D of G with k distinct elements such that the expressions \(xy^{-1}\) for all distinct two elements \(x,y\in D\), represent each non-identity element in G exactly \(\lambda \) times. A \((v,k,\lambda )\)-signed difference set is a generalization of a \((v,k,\lambda )\)-difference set D, which satisfies all properties of D, but has a sign for each element in D. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes.

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. Arasu K., Hollon J.R.: Group developed weighing matrices. Australas. J Comb. 55, 205–234 (2013).

    MathSciNet  Google Scholar 

  2. Arasu K., Jungnickel D., Ma S.L., Pott A.: Strongly regular Cayley graphs with \(\lambda - \mu =-1\). J. Combin. Theory Ser. A 67(1), 116–125 (1994).

    Article  MathSciNet  Google Scholar 

  3. Arasu K., Jungnickel D., Pott A.: Symmetric divisible designs with \(k-\lambda _1= 1\). Discret. Math. 97(1–3), 25–38 (1991).

    Article  Google Scholar 

  4. Balmaceda J.M.P., Estrella B.M.: Difference sets from unions of cyclotomic classes of orders \(12\), \(20\), and \(24\). Philippine J. Sci. 150(6B), 1803–1810 (2021).

    Article  Google Scholar 

  5. Brouwer A.E.: Parameters of strongly regular graphs (2019). https://www.win.tue.nl/~aeb/graphs/srg/srgtab.html.

  6. Chan Y.-K., Siu M.-K., Tong P.: Two-dimensional binary arrays with good autocorrelation. Inf. Control 42(2), 125–130 (1979).

    Article  MathSciNet  Google Scholar 

  7. Cusick T., Ding C., Renvall A.: Stream Ciphers and Number Theory, revised North Holland, Amsterdam (2005).

    Google Scholar 

  8. Ding C.: Codes from Difference Sets. World Scientific, Singapore (2014).

    Book  Google Scholar 

  9. Golay M.J.: Notes on digital coding. Proc. IEEE 37, 657 (1949).

    MathSciNet  Google Scholar 

  10. Gordon, D.M.: La Jolla Combinatorics Repository (2022). https://www.dmgordon.org.

  11. Gordon D.M.: Signed difference sets. Des. Codes Cryptogr. 91(5), 2107–2115 (2023). https://doi.org/10.1007/s10623-022-01171-8.

    Article  MathSciNet  Google Scholar 

  12. Hall M.: A survey of difference sets. Proc. Am. Math. Soc. 7(6), 975–986 (1956).

    Article  MathSciNet  Google Scholar 

  13. Helleseth T., Gong G.: New nonbinary sequences with ideal two-level autocorrelation. IEEE Trans. Inf. Theory 48(11), 2868–2872 (2002).

    Article  MathSciNet  Google Scholar 

  14. Katre S., Rajwade A.: Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order \(4\) and the corresponding Jacobsthal sum. Math. Scand. 60, 52–62 (1987).

    Article  MathSciNet  Google Scholar 

  15. Ko H.-P., Ray-Chaudhuri D.K.: Multiplier theorems. J. Combin. Theory Ser. A 30(2), 134–157 (1981).

    Article  MathSciNet  Google Scholar 

  16. Lebesgue V.A.: Sur l’impossibilité en nombres entiers de l’équation \(x^m=y^2+ 1\). Nouvelles Ann. Math. 9(9), 178–181 (1850).

    Google Scholar 

  17. Lehmer E.: On residue difference sets. Can. J. Math. 5, 425–432 (1953).

    Article  MathSciNet  Google Scholar 

  18. Leung K.H., Schmidt B.: Structure of group invariant weighing matrices of small weight. J. Combin. Theory Ser. A 154, 114–128 (2018).

    Article  MathSciNet  Google Scholar 

  19. Liu Z., Guan Y.L., Parampalli U., Hu S.: Spectrally-constrained sequences: bounds and constructions. IEEE Trans. Inf. Theory 64(4), 2571–2582 (2018).

    Article  MathSciNet  Google Scholar 

  20. Ma S.: Partial difference sets. Discret. Math. 52(1), 75–89 (1984).

    Article  MathSciNet  Google Scholar 

  21. Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4(4), 221–261 (1994).

    Article  MathSciNet  Google Scholar 

  22. Paley R.E.: On orthogonal matrices. J. Math. Phys. 12(1–4), 311–320 (1933).

    Article  Google Scholar 

  23. Polhill J.: Paley partial difference sets in groups of order \(n^4\) and \(9n^4\) for any odd \(n>1\). J. Combin. Theory Ser. A 117(8), 1027–1036 (2010).

    Article  MathSciNet  Google Scholar 

  24. Tan M.M.: Group invariant weighing matrices. Des. Codes Cryptogr. 86, 2677–2702 (2018).

    Article  MathSciNet  Google Scholar 

  25. Tao R., Feng T., Li W.: A construction of minimal linear codes from partial difference sets. IEEE Trans. Inf. Theory 67(6), 3724–3734 (2021).

    Article  MathSciNet  Google Scholar 

  26. Tsai L.-S., Chung W.-H., Shiu D.-S.: Synthesizing low autocorrelation and low PAPR OFDM sequences under spectral constraints through convex optimization and GS algorithm. IEEE Trans. Signal Process. 59(5), 2234–2243 (2011).

    Article  MathSciNet  Google Scholar 

  27. Wang Z.: Paley type partial difference sets in abelian groups. J. Combin. Des. 28(2), 149–152 (2020).

    Article  MathSciNet  Google Scholar 

  28. Wang Z., Gong G.: Constructions of complementary sequence sets and complete complementary codes by ideal two-level autocorrelation sequences and permutation polynomials. IEEE Trans. Inf. Theory (2023). https://doi.org/10.1109/TIT.2023.3258180.

    Article  MathSciNet  Google Scholar 

  29. Xiang Q.: On balanced binary sequences with two-level autocorrelation functions. IEEE Trans. Inf. Theory 44(7), 3153–3156 (1998).

    Article  MathSciNet  Google Scholar 

  30. Ye Z., Zhou Z., Liu Z., Tang X., Fan P.: New spectrally constrained sequence sets with optimal periodic cross-correlation. IEEE Trans. Inf. Theory 69(1), 610–625 (2022).

    Article  MathSciNet  Google Scholar 

  31. Ye Z., Zhou Z., Zhang S., Tang X.: Optimal ternary sequence sets rigorously achieving the Welch bound. Sci Sin Math 53, 1–12 (2023) (In Chinese).

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewer for providing helpful suggestions that directed them towards Refs. [15] and [16]. Furthermore, the authors extend their thanks to the associate editor and the anonymous reviewers for valuable comments that have significantly enhanced the quality of this paper. This project was supported by the National Key Research and Development Program of China under Grant 2020YFA0712100 and Grant 2018YFA0704703, the National Natural Science Foundation of China under Grant 11971325, Grant 12231014, Grant 12301429 and Grant 12301430, Beijing Scholars Program, and the Zhejiang Provincial Natural Science Foundation of China under Grant LQ23A010015.

Author information

Authors and Affiliations

  1. Zhejiang Lab, Hangzhou, 311100, China

    Zhiwen He

  2. Institute of Mathematics and Interdisciplinary Sciences, Xidian University, Xi’an, 710071, China

    Tingting Chen

  3. School of Mathematics Sciences, Capital Normal University, Beijing, 100048, China

    Gennian Ge

Authors
  1. Zhiwen He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tingting Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gennian Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tingting Chen.

Additional information

Communicated by K. T. Arasu.

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

He, Z., Chen, T. & Ge, G. New constructions of signed difference sets. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01389-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10623-024-01389-8

Keywords

Mathematics Subject Classification

Search

Navigation