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Abelian and non-abelian Paley type group schemes

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Abstract

In this paper, we present a construction of abelian Paley type group schemes which are inequivalent to Paley group schemes. We then determine the equivalence amongst their configurations, the Hadamard designs or the Paley type strongly regular graphs obtained from these group schemes, up to isomorphism. We also give constructions of several families of non-abelian Paley type group schemes using strong multiplier groups of the abelian Paley type group schemes, and present the first family of p-groups of non-square order and of non-prime exponent that contain Paley type group schemes for all odd primes p.

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References

  1. Beth T., Jungnickel D., Lenz H.: Design Theory, vol. 1, 2nd edn. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  2. Bierbrauer J.: New semifields, PN and APN functions. Des. Codes Cryptogr. 54, 189–200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bruck R.H.: Difference sets in a finite group. Trans. Am. Math. Soc. 78, 464–481 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  4. Camion P., Mann H.B.: Antisymmetric difference sets. J. Number Theory 4, 266–268 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen Y.Q.: Multiplicative characterization of some difference sets in elementary abelian groups. J. Comb. Inf. Syst. Sci. 34, 95–111 (2009)

    Google Scholar 

  6. Chen Y.Q.: Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles. J. Comb. Theory Ser. A 118(, 2185–2206), 118 , 2185–2206 (2011)

    Google Scholar 

  7. Chen Y.Q., Polhill J.: Paley type group schemes and planar Dembowski-Ostrom polynomials. Discret. Math. 311, 1349–1364 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen Y.Q., Xiang Q., Sehgal S.K.: An exponent bound on skew Hadamard abelian difference sets. Des. Codes Cryptogr. 4, 313–317 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davis J.A.: Partial difference sets in p-groups. Arch. Math. 63, 103–110 (1994)

    Article  MATH  Google Scholar 

  10. Ding C., Wang Z., Xiang Q.: Skew Hadamard difference sets from the Ree-Tits slice sympletic spreads in PG(3, 32h+1). J. Comb. Theory Ser. A 114, 867–887 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding C., Yin J.: A family of skewHadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006)

    Article  MATH  Google Scholar 

  12. Feng T.: Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism. J. Comb. Theory Ser. A 118, 27–36 (2011)

    Article  MATH  Google Scholar 

  13. Gao S., Wei W.D.: On nonabelian group difference sets. Discret. Math. 112, 93–102 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Johnson E.C.: Skew-Hadamardabelian group difference sets. J. Algebra 4, 388–402 (1966)

    Article  MathSciNet  Google Scholar 

  15. Kantor W.M.: 2-transitive symmetric designs. Trans. Am. Math. Soc. 146, 1–28 (1969)

    MathSciNet  MATH  Google Scholar 

  16. Kibler R.E.: A summary of noncyclic difference sets. k < 20. J. Comb. Theory Ser. A 25, 62–67 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leung K.H., Ma S.L.: Partial difference sets with Paley parameters. Bull. Lond. Math. Soc. 27, 553–564 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Muzychuk M.: On skew Hadamard difference sets, arXiv:1012.2089v1. http://arxiv.org/pdf/1012.2089.pdf.

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

    Google Scholar 

  21. Polhill J.: Paleytype partial difference sets in non p-groups. Des. Codes Cryptogr. 52, 163–169 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Turyn R.J.: Character sums and difference sets. Pac. J. Maths. 15, 319–346 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  24. Weng G., Hu L.: Some results on skew Hadamard difference sets. Des. Codes Cryptogr. 50, 93–105 (2009)

    Article  MathSciNet  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xiang Q.: Note on Paley Type Partial Difference Sets, in Groups, Difference Sets, and the Monster (Columbus, OH, 1993), pp. 239–244, Ohio State University Mathematical Research Institute Publications, 4, de Gruyter, Berlin (1996).

  27. Zha Z., Kyureghyan G.M., Wang X.: Perfect nonlinear binomials and their semifields. Finite Fields Appl. 15, 125–133 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Wright State University, Dayton, OH, 45435, USA

    Yu Qing Chen

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Tao Feng

Authors
  1. Yu Qing Chen
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  2. Tao Feng
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Correspondence to Yu Qing Chen.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.

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Chen, Y.Q., Feng, T. Abelian and non-abelian Paley type group schemes. Des. Codes Cryptogr. 68, 141–154 (2013). https://doi.org/10.1007/s10623-012-9640-3

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  • DOI: https://doi.org/10.1007/s10623-012-9640-3

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