Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 581–600 | Cite as

Paley type sets from cyclotomic classes and Arasu–Dillon–Player difference sets

  • Yu Qing ChenEmail author
  • Tao Feng


In this paper, we present constructions of abelian Paley type sets by using multiplicative characters of finite fields and Arasu–Dillon–Player difference sets. The constructions produce many new Paley type sets and their configurations that were previous unknown in our classification of Paley type sets in finite fields of small orders.


Difference set Paley set Paley type set Paley type partial difference set Skew Hadamard difference set Singer difference set 

Mathematics Subject Classification

05B10 05C25 05E18 05E30 



Y. Q. Chen would like to thank the Department of Mathematics at Zhejiang University for the hospitality he received during his visit when this research was initiated. The work of T. Feng was supported in part by the Fundamental Research Funds for the Central Universities, Zhejiang Provincial Natural Science Foundation.


  1. 1.
    Arasu K.T.: Sequences and arrays with desirable correlation properties.
  2. 2.
    Arasu K.T.: A reduction theorem for circulant weighing matrices. Australas. J. Comb. 18, 111–114 (1998).Google Scholar
  3. 3.
    Arasu K.T., Chen Y.Q., Dillon J.F., Liu X., Player K.J.: Abelian difference sets of order n dividing \(\lambda \). Des. Codes Cryptogr. 44, 307–319 (2007).Google Scholar
  4. 4.
    Arasu K.T., Dillon J.F., Player K.J.: Character sum factorizations yield perfect sequences (in press).Google Scholar
  5. 5.
    Arasu K.T., Leung K.H., Ma S.L., Nabavi A., Ray-Chaudhuri D.K.: Determination of all possible orders of weight 16 circulant weighing matrices. Finite Fields Appl. 12, 498–538 (2006).Google Scholar
  6. 6.
    Arasu K.T., Leung K.H., Ma S.L., Nabavi A., Ray-Chaudhuri D.K.: Circulant weighing matrices of weight \(2^{2t}\). Des. Codes Cryptogr. 41, 111–123 (2006).Google Scholar
  7. 7.
    Arasu K.T., Ma S.L.: Some new results on circulant weighing matrices. J. Algebraic Comb. 14, 91–101 (2001).Google Scholar
  8. 8.
    Berndt B.C., Evans R.J., Williams K.S.: Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1998).Google Scholar
  9. 9.
    Beth T., Jungnickel D., Lenz H.: Design Theory, vol. 1, 2nd edn. Cambridge University Press, Cambridge (1999).Google Scholar
  10. 10.
    Carlitz L.: A theorem on permutations in a finite field. Proc. Am. Math. Soc. 11, 456–459 (1960).Google Scholar
  11. 11.
    Camion P., Mann H.B.: Antisymmetric difference sets. J. Number Theory 4, 266–268 (1972).Google Scholar
  12. 12.
    Chen Y.Q.: On the existence of abelian Hadamard difference sets and a new family of difference sets. Finite Fields Appl. 3, 234–256 (1997).Google Scholar
  13. 13.
    Chen Y.Q.: Multiplicative characterization of some difference sets in elementary abelian groups. J. Comb. Inf. Syst. Sci. 34, 95–111 (2009).Google Scholar
  14. 14.
    Chen Y.Q.: Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles. J. Comb. Theory Ser. A 118, 2185–2206 (2011).Google Scholar
  15. 15.
    Chen Y.Q., Feng T.: Abelian and non-abelian Paley type group schemes. Des. Codes Cryptogr. 68, 141–154 (2013).Google Scholar
  16. 16.
    Chen Y.Q., Polhill J.: Paley type group schemes and planar Dembowski–Ostrom polynomials. Discret. Math. 311, 1349–1364 (2011).Google Scholar
  17. 17.
    Chen Y.Q., Xiang Q., Sehgal S.K.: An exponent bound on skew Hadamard abelian difference sets. Des. Codes Cryptogr. 4, 313–317 (1994).Google Scholar
  18. 18.
    Coulter R., Kosick P.: Commutative semifields of order 243 and 3125. Finite Fields Theory appl. Contemp. Math. 518, 129–136 (2010).Google Scholar
  19. 19.
    Davis J.A.: Partial difference sets in p-groups. Arch. Math. 63, 103–110 (1994).Google Scholar
  20. 20.
    Dillon J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974).Google Scholar
  21. 21.
    Dillon J.F.: Elementary Hadamard difference sets. In: Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (1975), pp. 237–249. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Manitoba (1975).Google Scholar
  22. 22.
    Dillon J.F.: Multiplicative difference sets via additive characters. Des. Codes Croptogr. 17, 225–235 (1999).Google Scholar
  23. 23.
    Ding C., Wang Z., Xiang Q.: Skew Hadamard difference sets from the Ree–Tits slice sympletic spreads in PG \((3,3^{2h+1})\). J. Comb. Theory Ser. A 114, 867–887 (2007).Google Scholar
  24. 24.
    Ding C., Yin J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006).Google Scholar
  25. 25.
    Feng T.: Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism. J. Comb. Theory Ser. A 118, 27–36 (2011).Google Scholar
  26. 26.
    Feng T., Momihara K., Xiang Q.: Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes, arXiv:1206.3354.Google Scholar
  27. 27.
    Feng T., Xiang Q.: Cyclotomic constructions of skew Hadamard difference sets. J. Comb. Theory Ser. A 119, 245–256 (2012).Google Scholar
  28. 28.
    Gordon B., Mills W.H., Welch L.R.: Some new difference sets. Can. J. Math. 14, 614–625 (1962).Google Scholar
  29. 29.
    Johnson E.C.: Skew-Hadamard abelian group difference sets. J. Algebra 4, 388–402 (1966).Google Scholar
  30. 30.
    Kantor W.M.: 2-Transitive symmetric designs. Trans. Am. Math. Soc. 146, 1–28 (1969).Google Scholar
  31. 31.
    Langevin P.: Calcus de certaines sommes de Gauss. J. Number Theory 63, 59–64 (1997).Google Scholar
  32. 32.
    Leung K.H., Ma S.L., Schmidt B.: Constructions of relative difference sets with classical parameters and circulant weighing matrices. J. Comb. Theory Ser. A 99, 111–127 (2002).Google Scholar
  33. 33.
    Lubotzky A., Phillips R., Sarnak P.: Ramanujan graphs. Combinatorica 8, 261–277 (1988).Google Scholar
  34. 34.
    Ma S.L.: Partial difference sets. Discret. Math. 52, 75–89 (1984).Google Scholar
  35. 35.
    Ma S.L.: Polynomial addition sets and symmetric difference sets. In: Ray-Chandhuri, D. (ed.) Coding Theory and Design Theory Part II: Design Theory, pp. 273–279. Springer, New York (1990).Google Scholar
  36. 36.
    Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4, 221–261 (1994).Google Scholar
  37. 37.
    Ma S.L.: Reversible relative difference sets. Combinatorica 12, 425–432 (1992).Google Scholar
  38. 38.
    Momihara K.: Skew Hadamard difference sets from cyclotomic strongly regular graphs. arXiv:1211. 2864v1.Google Scholar
  39. 39.
    Muzychuk M.: On skew Hadamard difference sets. arXiv:1012.2089v1.Google Scholar
  40. 40.
    Paley R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933).Google Scholar
  41. 41.
    Peisert W.: All self-complementary symmetric graphs. J. Algebra 240, 209–229 (2001).Google Scholar
  42. 42.
    Polhill J.: Paley type partial difference sets in non \(p\)-groups. Des. Codes Cryptogr. 52, 163–169 (2009).Google Scholar
  43. 43.
    Polhill J.: Paley type partial difference sets in groups of order \(n^4\) and \(9n^4\) for any odd \(n\). J. Comb. Theory Ser. A 117, 1027–1036 (2010).Google Scholar
  44. 44.
    Pott A.: Finite geometry and character theory. Lecture Notes in Mathematics, vol. 1601. Springer, Berlin, (1995).Google Scholar
  45. 45.
    Weng G., Qiu W., Wang Z., Xiang Q.: Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. Des. Codes Cryptogr. 44, 49–62 (2007).Google Scholar
  46. 46.
    Xiang Q.: Note on Paley type partial difference sets. Groups, Difference Sets, and the Monster (Columbus, OH, 1993), pp. 239–244. Ohio State University Mathematical Research Institute Publications, Berlin (1996).Google Scholar
  47. 47.
    Yamamoto K.: On congruences arising from relative Gauss sum. Number Theory and Combinatorics, pp. 423–446. World Scientific, Singapore (1955).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsWright State UniversityDaytonUSA
  2. 2.Department of MathematicsZhejiang UniversityHangzhouChina

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