Abstract
In this paper we show that starting from a symplectic semifield spread \({\mathcal{S}}\) of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of \({\mathcal{S}}\), and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) (q = s 2, s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125–133, 2009).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albert A.A.: Finite division algebras and finite planes. Proc. Symp. Appl. Math. 10, 53–70 (1960)
Albert A.A.: Generalized twisted fields. Pac. J. Math 11, 1–8 (1961)
Bader L., Kantor W.M., Lunardon G.: Symplectic spreads from twisted fields. Boll. Un. Math. Ital. 8, 383–389 (1994)
Biliotti M., Jha V., Johnson N.L.: Symplectic flock spreads in PG(3, q). Note Math. 24(1), 85–109 (2005)
Budaghyan, L., Helleseth, T.: New Perfect Nonlinear Multinomials over \({\mathbb{F}_{p^{2k}}}\) for any odd prime p. Lecture Notes Comput. Sci. 5203, 403–414, SETA (2008)
Cardinali I., Polverino O., Trombetti R.: Semifield planes of order q 4 with kernel \({\mathbb{F}_{q^2}}\) and center \({\mathbb{F}_q}\). Eur. J. Combin. 27, 940–961 (2006)
Coulter R.S., Henderson M.: Commutative presemifields and semifields. Adv. Math. 217, 282–304 (2008)
Coulter R.S., Henderson M., Kosick P.: Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptogr. 44, 275–286 (2007)
Coulter R.S., Matthews R.W.: Planar functions and planes of Lenz–Barlotti clas II. Des. Codes Cryptogr. 10, 167–184 (1997)
Dembowski P.: Finite Geometries. Springer, Berlin (1968)
Ding C., Yuang J.: A new family of skew Paley–Hadamard difference sets. J. Combin. Theory, Ser. A 113, 1526–1535 (2006)
Ebert G.L., Marino G., Polverino O., Trombetti R.: Infinite families of new semifields. Combinatorica 6, 637–663 (2009)
Ganley M.J.: Central weak nucleus semifields. Eur. J. Combin. 2, 39–347 (1981)
Harris J.: Algebraic Geometry, A First Course. Springer, New York (1992)
Hirschfeld J.W.P., Thas J.A.: General Galois geometries. Oxford University Press, Oxford (1991)
Johnson N.L., Jha V., Biliotti M.: Handbook of finite translation planes. In: Pure and Applied Mathematics (Boca Raton), vol. 289. Chapman & Hall/CRC, Boca Raton (2007)
Johnson N.L., Marino G., Polverino O., Trombetti R.: Semifields of order q 6 with left nucleus \({\mathbb{F}_{q^3}}\) and center \({\mathbb{F}_q}\). Finite Fields Appl. 14(2), 456–469 (2008)
Kantor W.M.: Commutative semifields and symplectic spreads. J. Algebra 270, 96–114 (2003)
Kantor W.M.: Isomorphisms of symplectic planes. Adv. Geom. 7, 553–557 (2007)
Kantor W.M., Williams M.E.: Symplectic semifield planes and \({\mathbb{Z}_4}\)-linear codes. Trans. Am. Math. Soc. 356(3), 895–938 (2004)
Knuth D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965)
Lavrauw M.: On the isotopism classes of finite semifields. Finite Fields Appl. 14(4), 897–910 (2008)
Lavrauw, M.: Finite semifields with a large nucleus and higher secant varieties to Segre varieties. Adv. Geom. (in press)
Lavrauw, M., Polverino, O.: Finite Semifields. In: Current Research Topics in Galois Geometry (to appear) (Nova Collected Works)
Lidl, R., Niederreiter, H.: Finite fields. In: Encyclopedia of Mathematics and its Applications, vol. 20. Addison-Wesley (now distributed by Cambridge University Press) (1983)
Lunardon G.: Translation ovoids. J. Geom. 76, 200–215 (2003)
Lunardon G.: Symplectic spreads and finite semifields. Des. Codes Cryptogr. 44, 39–48 (2007)
Lunardon G., Marino G., Polverino O., Trombetti R.: Translation dual of a semifield. J. Combin. Theory Ser. A 115, 1321–1332 (2008)
Lüneburg, H.: Translation planes. Springer, (1980)
Marino G., Polverino O., Trombetti R.: On \({{\mathbb F}_q}\)-linear sets of PG(3, q 3) and semifields. J. Combin. Theory Ser. A 114, 769–788 (2007)
Marino G., Polverino O., Trombetti R.: On semifields of type (q 2n, q n, q 2, q 2, q), n odd. Innov. Incidence Geom. 6(7), 209–227 (2009)
Maschietti A.: Symplectic translation planes. Lecture Notes Semin. Interdiscip. Math. II, 101–148 (2003)
Penttila T., Williams B.: Ovoids of parabolic spaces. Geom. Dedicata 82, 1–19 (2000)
Polverino O.: Linear sets in finite projective spaces. Discrete Math. 310, 3096–3107 (2010). doi:10.1016/j.disc.2009.04.007
Taylor D.E.: The geometry of the classical groups. Heldermann, Berlin (1992)
Thas J.A., Payne S.E.: Spreads and ovoids in finite generalized quadrangles. Geom. Dedicata 52(3), 227–253 (1994)
Zha Z., Kyureghyan G.M., Wang X.: Perfect nonlinear binomials and their semifields. Finite Fields Appl. 15(2), 125–133 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Editor in Chief.
This work was supported by the Research Project of MIUR (Italian Ministry for University and Research) and by the Research group GNSAGA of INDAM.
Rights and permissions
About this article
Cite this article
Lunardon, G., Marino, G., Polverino, O. et al. Symplectic semifield spreads of PG(5, q) and the veronese surface. Ricerche mat. 60, 125–142 (2011). https://doi.org/10.1007/s11587-010-0098-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-010-0098-1