On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$

Vito Napolitano

doi:10.36890/iejg.1049258

Research Article

On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$

Year 2022, , 79 - 82, 30.04.2022

Vito Napolitano

https://doi.org/10.36890/iejg.1049258

Abstract

In this paper $(q^{2} + q + 1)$ -sets of points in $P G (3, q)$ of type $(m, n, r)$ with respect to planes are studied, and as a by-product for $q$ odd a characterization of quadratic cones is obtained.

Keywords

Projective spaces, intersection numbers, oval, quadratic cone

References

[1] Aguglia, A., Cossidente, A., Korchmáros, G.: On quasi–Hermitian varieties. J. Comb. Des. 20, 433–447 (2012).
[2] Calderbank, R., Kantor, W. M.: The geometry of two-weight codes. Bull. London Math. Soc. 18, 97-122 (1986).
[3] De Clerck, F., Hamilton, N., O’ Keefe, C. M., Penttila, T.: Quasi-quadrics and related structures. Aust. J. Combin. 22, 151–166 (2000).
[4] DeWinter, S. Schillewaert, J.: A note on quasi–Hermitian varieties and singular quasi–quadrics. Bull. Belg. Math. Soc. Simon Stevin 17, 911—918 (2010).
[5] Durante, N, Napolitano, V., Olanda, D.: Sets of type (q + 1; n) in PG(3; q). J. Geom. 107, 9–18 (2016). DOI: 10.1007/s00022–015–0271–5
[6] Napolitano, V.: A characterization of the Hermitian variety in finite 3-dimensional projective spaces. The Electronic Journal of Combinatorics. 22 (1), 1–22 (2015).
[7] Napolitano, V. : On quasi–Hermitian varieties in PG(3; q2). Discrete Math. 339, 511–514 (2016). DOI 10.1016/j.disc.2015.09.013
[8] Napolitano V.: Cones in PG(3; q). Note Mat. 40 ( 1), 81–86 (2020). DOI:10.1285/i15900932v40n1p81
[9] Segre, B.: Ovals in a finite projective plane. Canad. J. Math. 7, 414-–416 (1955).
[10] Zannetti, M.: A combinatorial characterization of parabolic quadrics, Journal of Discrete Mathematical Sciences and Cryptography 12 (6), 707-715 (2009). DOI: 10.1080/09720529.2009.10698266
[11] Zuanni F.: A characterization of the oval cone in PG(3; q), Journal of Discrete Mathematical Sciences and Cryptography 24 (3), 859–863 (2021). DOI 10.1080/09720529.2021.1895506

Year 2022, , 79 - 82, 30.04.2022

Vito Napolitano

https://doi.org/10.36890/iejg.1049258

Abstract

References

[1] Aguglia, A., Cossidente, A., Korchmáros, G.: On quasi–Hermitian varieties. J. Comb. Des. 20, 433–447 (2012).
[2] Calderbank, R., Kantor, W. M.: The geometry of two-weight codes. Bull. London Math. Soc. 18, 97-122 (1986).
[3] De Clerck, F., Hamilton, N., O’ Keefe, C. M., Penttila, T.: Quasi-quadrics and related structures. Aust. J. Combin. 22, 151–166 (2000).
[4] DeWinter, S. Schillewaert, J.: A note on quasi–Hermitian varieties and singular quasi–quadrics. Bull. Belg. Math. Soc. Simon Stevin 17, 911—918 (2010).
[5] Durante, N, Napolitano, V., Olanda, D.: Sets of type (q + 1; n) in PG(3; q). J. Geom. 107, 9–18 (2016). DOI: 10.1007/s00022–015–0271–5
[6] Napolitano, V.: A characterization of the Hermitian variety in finite 3-dimensional projective spaces. The Electronic Journal of Combinatorics. 22 (1), 1–22 (2015).
[7] Napolitano, V. : On quasi–Hermitian varieties in PG(3; q2). Discrete Math. 339, 511–514 (2016). DOI 10.1016/j.disc.2015.09.013
[8] Napolitano V.: Cones in PG(3; q). Note Mat. 40 ( 1), 81–86 (2020). DOI:10.1285/i15900932v40n1p81
[9] Segre, B.: Ovals in a finite projective plane. Canad. J. Math. 7, 414-–416 (1955).
[10] Zannetti, M.: A combinatorial characterization of parabolic quadrics, Journal of Discrete Mathematical Sciences and Cryptography 12 (6), 707-715 (2009). DOI: 10.1080/09720529.2009.10698266
[11] Zuanni F.: A characterization of the oval cone in PG(3; q), Journal of Discrete Mathematical Sciences and Cryptography 24 (3), 859–863 (2021). DOI 10.1080/09720529.2021.1895506

There are 11 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Article
Authors	Vito Napolitano 0000-0002-2504-6967
Publication Date	April 30, 2022
Acceptance Date	April 8, 2022
Published in Issue	Year 2022

Cite

APA	Napolitano, V. (2022). On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$. International Electronic Journal of Geometry, 15(1), 79-82. https://doi.org/10.36890/iejg.1049258
AMA	Napolitano V. On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$. Int. Electron. J. Geom. April 2022;15(1):79-82. doi:10.36890/iejg.1049258
Chicago	Napolitano, Vito. “On $(q^2+q+1)$-Sets of Plane-Type $(m, N, r)_2$ in $\mathrm{PG}(3, q)$”. International Electronic Journal of Geometry 15, no. 1 (April 2022): 79-82. https://doi.org/10.36890/iejg.1049258.
EndNote	Napolitano V (April 1, 2022) On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$. International Electronic Journal of Geometry 15 1 79–82.
IEEE	V. Napolitano, “On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$”, Int. Electron. J. Geom., vol. 15, no. 1, pp. 79–82, 2022, doi: 10.36890/iejg.1049258.
ISNAD	Napolitano, Vito. “On $(q^2+q+1)$-Sets of Plane-Type $(m, N, r)_2$ in $\mathrm{PG}(3, q)$”. International Electronic Journal of Geometry 15/1 (April 2022), 79-82. https://doi.org/10.36890/iejg.1049258.
JAMA	Napolitano V. On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$. Int. Electron. J. Geom. 2022;15:79–82.
MLA	Napolitano, Vito. “On $(q^2+q+1)$-Sets of Plane-Type $(m, N, r)_2$ in $\mathrm{PG}(3, q)$”. International Electronic Journal of Geometry, vol. 15, no. 1, 2022, pp. 79-82, doi:10.36890/iejg.1049258.
Vancouver	Napolitano V. On $(q^2+q+1)$-Sets of Plane-Type $(m, n, r)_2$ in $\mathrm{PG}(3, q)$. Int. Electron. J. Geom. 2022;15(1):79-82.