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A New Infinite Family of Hemisystems of the Hermitian Surface

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Abstract

In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q2). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q2) admitting \(C_{\left( {q^3 + 1} \right)/4} :C_3 \).

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References

  1. J. Bamberg, M. Giudici and G. F. Royle: Every flock generalized quadrangle has a hemisystem, Bull. Lond. Math. Soc. 42 (2010), 795–810.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Bamberg, M. Giudici and G. F. Royle: Hemisystems of small flock generalized quadrangles, Des. Codes Cryptogr. 67 (2013), 137–157.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Bamberg, S. Kelly, M. Law and T. Penttila: Tight sets and m-ovoids of finite polar spaces, J. Combin. Theory Ser. A 114 (2007), 1293–1314.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Bamberg and T. Penttila: Overgroups of cyclic Sylow subgroups of linear groups, Comm. Algebra 36 (2008), 2503–2543.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Bannai, O. Shimabukuro and H. Tanaka: Finite Euclidean graphs and Ramanujan graphs, Discrete Mathematics 309 (2009), 6126–6134.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. C. Berndt, R. J. Evans and K. S. Williams: Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1998, A Wiley-Interscience Publication.

    MATH  Google Scholar 

  7. A. A. Bruen and J. W. P. Hirschfeld: Applications of line geometry over finite fields. II. The Hermitian surface, Geom. Dedicata 7 (1978), 333–353.

    MathSciNet  MATH  Google Scholar 

  8. A. Cossidente and T. Penttila: Hemisystems on the Hermitian surface, J. London Math. Soc. 72 (2005), 731–741.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. De Beule, J. Demeyer, K. Metsch and M. Rodgers: A new family of tight sets in Q+(5,q), Des. Codes Cryptogr. 78 (2016), 655–678.

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Feng, K. Momihara and Q. Xiang: Cameron-Liebler line classes with parameter \(x = \frac{q^{2}-1}{2}\), J. Combin. Theory Ser. A 133 (2015), 307–338.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. W. P. Hirschfeld: Projective geometries over finite fields, The Clarendon Press, Oxford University Press, New York, 1979, Oxford Mathematical Monographs.

    MATH  Google Scholar 

  12. B. Segre: Forme e geometrie hermitiane, con particolare riguardo al caso finito, Ann. Mat. Pura Appl. 70 (1965), 1–201.

    Article  MATH  Google Scholar 

  13. J. A. Thas: Projective geometry over a finite field, in: Handbook of incidence geometry, 295–347, North-Holland, Amsterdam, 1995.

    Chapter  Google Scholar 

  14. E. R. van Dam, W. J. Martin and M. Muzychuk: Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems, J. Combin. Theory Ser. A 120 (2013), 1401–1439.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Yamamoto and M. Yamada: Williamson Hadamard matrices and Gauss sums, J. Math. Soc. Japan 37 (1985), 703–717.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, W.A., 6009, Australia

    John Bamberg & Melissa Lee

  2. Department of Mathematics, Faculty of Education, Kumamoto University, 2-40-1 Kurokami, Kumamoto, 860-8555, Japan

    Koji Momihara

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

    Qing Xiang

Authors
  1. John Bamberg
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  2. Melissa Lee
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  3. Koji Momihara
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  4. Qing Xiang
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Corresponding author

Correspondence to Qing Xiang.

Additional information

The first author acknowledges the support of the Australian Research Council Future Fellowship FT120100036.

The second author acknowledges the support of a Hackett Postgraduate Research Scholarship.

The third author acknowledges the support by JSPS under Grant-in-Aid for Young Scientists (B) 25800093 and Scientific Research (B) 15H03636.

The fourth author acknowledges the support of an NSF grant DMS-1600850.

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Bamberg, J., Lee, M., Momihara, K. et al. A New Infinite Family of Hemisystems of the Hermitian Surface. Combinatorica 38, 43–66 (2018). https://doi.org/10.1007/s00493-016-3525-4

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  • DOI: https://doi.org/10.1007/s00493-016-3525-4

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