Ramanujan graphs and expander families constructed from p-ary bent functions | Designs, Codes and Cryptography Skip to main content
Springer Nature Link
Log in

Ramanujan graphs and expander families constructed from p-ary bent functions

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

Abstract

We present a method for constructing an infinite family of non-bipartite Ramanujan graphs. We mainly employ p-ary bent functions of \((p-1)\)-form for this construction, where p is a prime number. Our result leads to construction of infinite families of expander graphs; this is due to the fact that Ramanujan graphs play as base expanders for constructing further expanders. For our construction we directly compute the eigenvalues of the Ramanujan graphs arsing from p-ary bent functions. Furthermore, we establish a criterion on the regularity of p-ary bent functions in m variables of \((p-1)\)-form when m is even. Finally, using weakly regular p-ary bent functions of \(\ell \)-form, we find (amorphic) association schemes in a constructive way; this resolves the open case that \(\ell = p-1\) for \(p >2\) for finding (amorphic) association schemes.

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
¥17,985 /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 includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Alon N.: Eigenvalues and expanders. Combinatorica 6(2), 83–96 (1986).

    Article  MathSciNet  Google Scholar 

  2. Arias de Reyna J.: Finite fields and Ramanujan graphs. J. Combin. Theory Ser. B 70(2), 259–264 (1997).

    Article  MathSciNet  Google Scholar 

  3. Bibak K., Kapron B.M., Srinivasan V.: The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs. IEEE Trans. Inf. Theory 62(11), 6355–6358 (2016).

    Article  MathSciNet  Google Scholar 

  4. Brouwer A.: Web database of strongly regular graphs. https://www.win.tue.nl/~aeb/graphs/srg/srgtab.html.

  5. Chee Y.M., Tan Y., Zhang X.D.: Strongly regular graphs constructed from \(p\)-ary bent functions. J. Algebr. Combin. 34(2), 251–266 (2011).

    Article  MathSciNet  Google Scholar 

  6. Chiu P.: Cubic Ramanujan graphs. Combinatorica 12, 275–285 (1992).

    Article  MathSciNet  Google Scholar 

  7. Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007).

    Google Scholar 

  8. Davidoff G., Sarnak P., Valette A.: Elementary Number Theory, Group Theory, and Ramanujan Graphs. Cambridge University Press, Cambridge (2003).

    MATH  Google Scholar 

  9. Godsil C.D.: Algebraic Combinatorics. Chapman & Hall/CRC, Boca Raton (1993).

    MATH  Google Scholar 

  10. Goldreich O., Impagliazzo R., Levin L., Venkatesan R., Zuckerman D.: Security preserving amplification of hardness. In: 31st Annual Symposium on Foundations of Computer Science, vol. I (1990), IEEE Computer Society Press, Los Alamitos, CA, pp. 318–326. Proofs of two conjectures on ternary weakly regular bent functions. IEEE Trans. Inf. Theory 55(11), 5272–5283 (2009).

  11. Hoory S., Linial N., Wigderson A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–561 (2006).

    Article  MathSciNet  Google Scholar 

  12. Hyun J.Y., Lee Y.: Characterization of \(p\)-ary bent functions in terms of strongly regular graphs. IEEE Trans. Inf. Theory 65(1), 676–684 (2019).

    Article  MathSciNet  Google Scholar 

  13. Kalton N.J., Roberts J.W.: Uniformly exhaustive submeasures and nearly additive set functions. Trans. Am. Math. Soc. 278(2), 803–816 (1983).

    Article  MathSciNet  Google Scholar 

  14. Krebs M., Shaheen A.: Expander families and Cayley Graphs: A Beginner’s Guide. Oxford University Press, Oxford (2011).

    MATH  Google Scholar 

  15. Kumar P.V., Scholtz R.A., Welch L.R.: Generalized bent functions and their properties. J. Combin. Theory Ser. A 40(1), 90–107 (1985).

    Article  MathSciNet  Google Scholar 

  16. Lubotzky A.: Expander graphs in pure and applied mathematics. Bull. Am. Math. Soc. 49(1), 113–162 (2012).

    Article  MathSciNet  Google Scholar 

  17. Lubotzky A., Phillips R., Sarnak P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988).

    Article  MathSciNet  Google Scholar 

  18. Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4(3), 221–261 (1994).

    Article  MathSciNet  Google Scholar 

  19. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1998).

    MATH  Google Scholar 

  20. Marcus A.W., Spielman D.A., Srivastava N.: Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. Math. (2) 182(1), 307–325 (2015).

    Article  MathSciNet  Google Scholar 

  21. Margulis G.A.: Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Probl. Inf. Transm. 24(1), 39–46 (1988).

    MathSciNet  MATH  Google Scholar 

  22. Mesnager S., Tang C., Qi Y.: 2-correcting Lee codes: (quasi)-perfect spectral conditions and some constructions. IEEE Trans. Inf. Theory 64(4), part 2, 3031–3041 (2018).

    Article  MathSciNet  Google Scholar 

  23. Morgenstern M.: Existence and explicit constructions of \(q+1\) regular Ramanujan graphs for every prime power \(q\). J. Combin. Theory Ser. B 62(1), 44–62 (1994).

    Article  MathSciNet  Google Scholar 

  24. Pippenger N.: Sorting and selecting in rounds. SIAM J. Comput. 16(6), 1032–1038 (1987).

    Article  MathSciNet  Google Scholar 

  25. Pott A., Tan Y., Feng T., Ling S.: Association schemes arising from bent functions. Des. Codes Cryptogr. 59(1–3), 319–331 (2011).

    Article  MathSciNet  Google Scholar 

  26. Reingold O., Vadhan S., Wigderson A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. Math. (2) 155(1), 157–187 (2002).

    Article  MathSciNet  Google Scholar 

  27. Sipser M., Spielman D.A.: Expander codes. IEEE Trans. Inf. Theory 42(6), part 1, 1710–1722 (1996).

    Article  MathSciNet  Google Scholar 

  28. Tan Y., Pott A., Feng T.: Strongly regular graphs associated with ternary bent functions. J. Combin. Theory Ser. A 117(6), 668–682 (2010).

    Article  MathSciNet  Google Scholar 

  29. Tang C., Li N., Qi Y., Zhou Z., Helleseth T.: Linear codes with two or three weights from weakly regular bent functions. IEEE Trans. Inf. Theory 62(3), 1166–1176 (2016).

    Article  MathSciNet  Google Scholar 

  30. van Dam E.R.: Strongly regular decompositions of the complete graph. J. Algebr. Combin. 17(2), 181–201 (2003).

    Article  MathSciNet  Google Scholar 

  31. Williamson C.: Spectral Graph Theory, Expanders, and Ramanujan Graphs. University of Washington (2014). https://sites.math.washington.edu/~morrow/papers/chris-thesis.pdf.

  32. Zinovev V.A., Ericson T.: Fourier-invariant pairs of partitions of finite abelian groups, and association schemes. Probl. Inf. Transm. 45(3), 221–231 (2009).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

J.Y. Hyun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1D1A1B05030707), J. Lee by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A6A3A11030486) and 2019 Research Grant from Kangwon National University, and Y. Lee by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A2B2004574). We express our gratitude to the reviewers for their very helpful comments, which lead to improvement of the exposition of this paper.

Author information

Authors and Affiliations

  1. Konkuk University, Glocal Campus, 268 Chungwon-daero, Chungju-si, Chungcheongbuk-do, 27478, South Korea

    Jong Yoon Hyun

  2. Department of Mathematics Education, Kangwon National University, Chuncheon-si, Gangwon-do, 24341, South Korea

    Jungyun Lee

  3. Department of Mathematics, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul, 03760, South Korea

    Yoonjin Lee

Authors
  1. Jong Yoon Hyun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jungyun Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoonjin Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoonjin Lee.

Additional information

Communicated by G. Kyureghyan.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hyun, J.Y., Lee, J. & Lee, Y. Ramanujan graphs and expander families constructed from p-ary bent functions. Des. Codes Cryptogr. 88, 453–470 (2020). https://doi.org/10.1007/s10623-019-00692-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-019-00692-z

Keywords

Mathematics Subject Classification

Search

Navigation