Skip to main content
SpringerLink
Log in

Cyclotomy, Gauss Sums, Difference Sets and Strongly Regular Cayley Graphs

  • Conference paper
Sequences and Their Applications – SETA 2012 (SETA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7280))

Included in the following conference series:

Abstract

We survey recent results on constructions of difference sets and strongly regular Cayley graphs by using union of cyclotomic classes of finite fields. Several open problems are raised.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baumert, L.D.: Cyclic Difference Sets. Lecture Notes in Mathematics, vol. 182. Springer (1971)

    Google Scholar 

  2. Baumert, L.D., Fredricksen, H.: The cyclotomic numbers of order eighteen with applications to difference sets. Math. Comp. 21, 204–219 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baumert, L.D., Mills, M.H., Ward, R.L.: Uniform Cyclotomy. J. Number Theory 14, 67–82 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. A Wiley-Interscience Publication (1998)

    Google Scholar 

  5. Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn. Encyclopedia of Mathematics and its Applications, 78, vol. I. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  6. Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer Universitext (2012)

    Google Scholar 

  7. Brouwer, A.E., Wilson, R.M., Xiang, Q.: Cyclotomy and strongly regular graphs. J. Algebraic Combin. 10, 25–28 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calderbank, R., Kantor, W.M.: The geometry of two-weight codes. Bull. London Math. Soc. 18, 97–122 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cameron, P.: Strongly regular graphs. In: Beineke, L.W., Wilson, R.J. (eds.) Topics in Algebraic Graph Theory, pp. 203–221. Cambridge Univ. Press, Cambridge (2004)

    Google Scholar 

  10. Evans, R.J.: Nonexistence of twentieth power residue difference sets. Acta Arith. 84, 397–402 (1999)

    Google Scholar 

  11. Feng, T., Xiang, Q.: Cyclotomic constructions of skew Hadamard difference sets. J. Combin. Theory (A) 119, 245–256 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feng, T., Xiang, Q.: Strongly regular graphs from unions of cyclotomic classes. J. Combin. Theory (B) (in press)

    Google Scholar 

  13. Feng, T., Wu, F., Xiang, Q.: Pseudocyclic and non-amorphic fusion schemes of the cyclotomic association schemes. Des. Codes and Cryptogr. (in press)

    Google Scholar 

  14. Feng, T., Momihara, K., Xiang, Q.: Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes. arXiv:1201.0701 (submitted)

    Google Scholar 

  15. Feng, K.Q., Yang, J., Luo, S.X.: Gauss sum of index 4: (1) cyclic case. Acta Math. Sin. (Engl. Ser.) 21, 1425–1434 (2005)

    Google Scholar 

  16. Ge, G., Xiang, Q., Yuan, T.: Constructions of strongly regular Cayley graphs using index four Gauss sums. arXiv:1201.0702 (submitted)

    Google Scholar 

  17. Godsil, C., Royle, G.: Algebraic Graph Theory. GTM, vol. 207. Springer (2001)

    Google Scholar 

  18. Hall Jr., M.: A survey of difference sets. Proc. Amer. Math. Soc. 7, 975–986 (1956)

    Article  MathSciNet  Google Scholar 

  19. Ikuta, T., Munemasa, A.: Pseudocyclic association schemes and strongly regular graphs. European J. Combin. 31, 1513–1519 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jungnickel, D.: Difference sets. In: Contemporary Design Theory. Wiley-Intersci. Ser. Discrete Math. Optim., pp. 241–324. Wiley, New York (1992)

    Google Scholar 

  21. Jungnickel, D., Schmidt, B.: Difference sets: an update. In: Geometry, Combinatorial Designs and Related Structures (Spetses, 1996). London Math. Soc. Lecture Note Ser., vol. 245, pp. 89–112. Cambridge Univ. Press, Cambridge (1997)

    Google Scholar 

  22. Jungnickel, D., Schmidt, B.: Difference sets: a second update. In: Combinatorics 1998 (Mondello). Rend. Circ. Mat. Palermo (2) (suppl. no. 53), pp. 89–118 (1998)

    Google Scholar 

  23. Lander, E.S.: Symmetric Designs: An Algebraic Approach. London Math. Society Lecture Note Series, vol. 74. Cambridge University Press (1983)

    Google Scholar 

  24. de Lange, C.L.M.: Some new cyclotomic strongly regular graphs. J. Alg. Combin. 4, 329–330 (1995)

    Article  MATH  Google Scholar 

  25. Langevin, P.: Calculs de certaines sommes de Gauss. J. Number Theory 63, 59–64 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Langevin, P.: A new class of two-weight codes. In: Cohen, S., Niederreiter, H. (eds.) Finite Fields and Applications (Glasgow 1995). London Math. Soc. Lecture Note Series, vol. 233, pp. 181–187. Cambridge University Press (1996)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Liebler, R.A.: Constructive representation theoretic methods and non-abelian difference sets. In: Difference Sets, Sequences and their Correlation Properties (Bad Windsheim 1998). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 542, pp. 331–352. Kluwer Acad. Publ., Dordrecht (1999)

    Google Scholar 

  29. van Lint, J.H., Schrijver, A.: Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields. Combinatorica 1, 63–73 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Mbodj, O.D.: Quadratic Gauss sums. Finite Fields and Appl. 4, 347–361 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. McEliece, R.J.: Irreducible cyclic codes and Gauss sums. In: Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974), Part 1: Theory of Designs, Finite Geometry and Coding Theory. Math. Centre Tracts, vol. 55, pp. 179–196. Math. Centrum, Amsterdam (1974)

    Google Scholar 

  33. Meijer, P., van der Vlugt, M.: The evaluation of Gauss sums for characters of 2-power order. J. Number Theory 100, 381–395 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Momihara, K.: Strongly regular Cayley graphs, skew Hadamard difference sets, and rationality of relative Gauss sums. arXiv:1202.6414

    Google Scholar 

  35. Muzychuk, M.: On T-amorphous association schemes (preprint)

    Google Scholar 

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

    Google Scholar 

  37. Schmidt, B., White, C.: All two-weight irreducible cyclic codes. Finite Fields Appl. 8, 1–17 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Storer, T.: Cyclotomy and difference sets. Markham, Chicago (1967)

    MATH  Google Scholar 

  39. Thas, K.: Finite flag-transitive projective planes: a survey and some remarks. Discrete Math. 266, 417–429 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Thas, K., Zagier, D.: Finite projective planes, Fermat curves, and Gaussian periods. J. Eur. Math. Soc. (JEMS) 10, 173–190 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu, F.: Constructions of strongly regular Cayley graphs using even index Gauss sums (preprint)

    Google Scholar 

  42. Xiang, Q.: Recent results on difference sets with classical parameters. In: Difference Sets, Sequences and their Correlation Properties (Bad Windsheim, 1998). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 542, pp. 419–437. Kluwer Acad. Publ., Dordrecht (1999)

    Google Scholar 

  43. Xiang, Q.: Recent progress in algebraic design theory. Finite Fields Appl. (Ten Year Anniversary Edition) 11, 622–653 (2005)

    Article  MATH  Google Scholar 

  44. Yang, J., Luo, S.X., Feng, K.Q.: Gauss sum of index 4:(2) Non-cyclic case. Acta Math. Sin. (Engl. Ser.) 22, 833–844 (2006)

    Google Scholar 

  45. Yang, J., Xia, L.: Complete solving of explicit evaluation of Gauss sums in the index 2 case. Sci. China Ser. A 53, 2525–2542 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Qing Xiang

Authors
  1. Qing Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, P.O. Box 7803, 5020, Bergen, Norway

    Tor Helleseth

  2. Department of Mathematics, Simon Fraser University, 8888 University Drive, V5A 1S6, Burnaby, BC, Canada

    Jonathan Jedwab

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Xiang, Q. (2012). Cyclotomy, Gauss Sums, Difference Sets and Strongly Regular Cayley Graphs. In: Helleseth, T., Jedwab, J. (eds) Sequences and Their Applications – SETA 2012. SETA 2012. Lecture Notes in Computer Science, vol 7280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30615-0_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-30615-0_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30614-3

  • Online ISBN: 978-3-642-30615-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation