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Classifying sets of class \([1,q+1,2q+1,q^2+q+1]_2\) in PG(r, q), \(r\ge 3\)

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Abstract

In this paper we remove the solid incidence assumption in the characterization of J. Schillewaert (A characterization of quadrics by intersection numbers, Des. Codes Cryptogr., 47 (2008), 165–175) by proving that quadric plane incidence numbers imply quadric solid incidence numbers, except for the dual complete 11-cap of PG(4, 3). Furthermore, new characterizations of the parabolic quadric Q(4, q) and the ovoidal cone of PG(4, q) are provided.

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References

  1. Barwick S.G., Hui A.M., Jackson W.A., Schillewaert J.: Characterising hyperbolic hyperplanes of a non-singular quadric in \(PG(4, q)\). Des. Codes Cryptogr. 88, 33–39 (2020). https://doi.org/10.1007/s10623-019-00669-y.

    Article  MathSciNet  MATH  Google Scholar 

  2. Ferri O., Tallini G.: A characterization of nonsingular quadrics in \(PG(4, q)\). Rend. Mat. Appl. 11(1), 15–21 (1991).

    MathSciNet  MATH  Google Scholar 

  3. Hirschfeld J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford University Press, New York (1985).

    MATH  Google Scholar 

  4. Hirschfeld J.W.P., Thas J.A.: General Galois Geometry. Springer Monographs in MathematicsSpringer, London (2016).

    Book  Google Scholar 

  5. S. Innamorati, F. Zuanni: On sets of class \([1,q+1,2q+1]_2\) in \(PG(3,q)\), Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali, 96 (S2), Article number 6. https://doi.org/10.1478/AAPP.96S2A6 (2018)

  6. Napolitano V.: Ruled sets of plane-type \((m, h)_2\) in \(PG(3, q)\) with \(m\le q\). J. Geome. 108, 953–960 (2017).

    Article  Google Scholar 

  7. Pellegrino G.: Sulle proprietà della 11-calotta completa di \(S_{4,3}\) e su un B.I.B.-disegno ad essa collegato. Boll. Un. Mat. Ital. 4(7), 463–470 (1983).

    MATH  Google Scholar 

  8. Schillewaert J.: A characterization of quadrics by intersection numbers. Des. Codes Cryptogr. 47, 165–175 (2008).

    Article  MathSciNet  Google Scholar 

  9. Tallini G.: Graphic characterization of algebraic varieties in a Galois space. Teorie Combinatorie, Atti dei Convegni Lincei 17(2), 153–165 (1976).

    MathSciNet  MATH  Google Scholar 

  10. Tallini Scafati M.: The \(k\)-sets of \(PG(r,q)\) from the character point of view. In: Baker C.A., Batten L.M. (eds.) Finite Geometries, pp. 321–326. Marcel Dekker Inc., New York (1985).

    Google Scholar 

  11. Thas J.A.: A combinatorial problem. Geom. Dedicata 1, 236–240 (1973).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are grateful to the three anonymous referees for their useful suggestions which greatly improved the presentation of this paper.

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Authors and Affiliations

  1. Department of Industrial and Information Engineering and Economics, University of L’Aquila, Piazzale Ernesto Pontieri, 1 (Monteluco di Roio), 67100, L’Aquila, Italy

    Stefano Innamorati & Fulvio Zuanni

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  1. Stefano Innamorati
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  2. Fulvio Zuanni
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Correspondence to Fulvio Zuanni.

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Communicated by K.Metsch.

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Dedicated to Prof. Osvaldo Ferri on the occasion of his 85th birthday.

Remembering Prof. Giuseppe Tallini on the 25th anniversary of his death.

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Innamorati, S., Zuanni, F. Classifying sets of class \([1,q+1,2q+1,q^2+q+1]_2\) in PG(r, q), \(r\ge 3\). Des. Codes Cryptogr. 89, 489–496 (2021). https://doi.org/10.1007/s10623-020-00833-9

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  • DOI: https://doi.org/10.1007/s10623-020-00833-9

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