Journal of Algebraic Combinatorics (EMIS Electronic Version)

Home > Vol 33, No 1 (2011)

Slices of the unitary spread

G. Lunardon , L. Parlato² , V. Pepe² and R. Trombetti⁴

²Department of Pure Mathematics and Computer Algebra, University of Gent, Krijgslaan 281, 9000 Gent, Belgium
⁴V. Pepe Department of Pure Mathematics and Computer Algebra, University of Gent, Krijgslaan 281, 9000 Gent, Belgium

DOI: 10.1007/s10801-010-0232-x

Abstract

We prove that slices of the unitary spread of Q ⁺(7, q) \mathcal{Q}^{+}(7,q), q\equiv 2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of PΓ O ⁺(8, q) fixing the unitary spread. When q is even, there is a connection between spreads of Q ⁺(7, q) \mathcal{Q}^{+}(7,q) and symplectic 2-spreads of PG(5, q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173-194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q ⁺(7, q) \mathcal{Q}^{+}(7,q), q=2 ^{2 h+1}. Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151-165, 1982. When q=3 ^h, we classify, up to the action of the stabilizer in PΓ O(7, q) of the unitary spread of Q(6, q), those among its slices producing spreads of the elliptic quadric Q ^-(5, q) \mathcal{Q}^{-}(5,q).

Pages: 37–56

Keywords: keywords ovoid; unitary spread; slice

Full Text: PDF

References

1. Baker, R.D., Ebert, G.L., Wantz, K.L.: Regular hyperbolic fibrations. Adv. Geom. 1(2), 119-144 (2001)
2. Cooperstein, B.N.: Hyperplane sections of Kantor's unitary ovoid. Des. Codes Cryptogr. 23(2), 185- 195 (2001)
3. Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. thesis, Univ. of Meryland, College Park (1974)
4. Dye, R.: Partitions and their stabilizers for line complexes and quadrics. Ann. Mat. Pura Appl. 114(4), 173-194 (1977)
5. Giuzzi, L.: Collineation groups of the intersection of two classical unitals. J. Comb. Des. 9(6), 445- 459 (2001)
6. Hirschfeld, J.W.P.: Finite Projective Spaces of Three Dimension. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1985)
7. Hirschfeld, J.W.P., Thas, J.A.: General Galois Geometries. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991)
8. Kantor, W.M.: Ovoids and translation planes. Can. J. Math. 34(5), 1195-1207 (1982)
9. Kantor, W.M.: Spreads, translation planes and Kerdock sets I. SIAM J. Algebr. Discrete Methods 3(2), 151-165 (1982)
10. Kestenband, B.C.: Unital intersections in finite projective planes. Geom. Dedicata 11(1), 107-117 (1981)
11. Lunardon, G.: Normal spreads. Geom. Dedicata 75(3), 245-261 (1999)
12. Moisio, M.: Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm. Acta Arith. 132(4), 329-350 (2008)
13. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. Pitman Advanced Publishing Program (1984) J Algebr Comb (2011) 33: 37-56

	EMIS Electronic Library of Mathematics (ELibM) The Open Access Repository of Mathematics
	JOURNAL OF ALGEBRAIC COMBINATORICS
	Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
	Home > Vol 33, No 1 (2011) Slices of the unitary spread G. Lunardon , L. Parlato² , V. Pepe² and R. Trombetti⁴ ²Department of Pure Mathematics and Computer Algebra, University of Gent, Krijgslaan 281, 9000 Gent, Belgium ⁴V. Pepe Department of Pure Mathematics and Computer Algebra, University of Gent, Krijgslaan 281, 9000 Gent, Belgium DOI: 10.1007/s10801-010-0232-x Abstract We prove that slices of the unitary spread of Q ⁺(7, q) \mathcal{Q}^{+}(7,q), q\equiv 2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of PΓ O ⁺(8, q) fixing the unitary spread. When q is even, there is a connection between spreads of Q ⁺(7, q) \mathcal{Q}^{+}(7,q) and symplectic 2-spreads of PG(5, q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173-194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q ⁺(7, q) \mathcal{Q}^{+}(7,q), q=2 ^{2 h+1}. Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151-165, 1982. When q=3 ^h, we classify, up to the action of the stabilizer in PΓ O(7, q) of the unitary spread of Q(6, q), those among its slices producing spreads of the elliptic quadric Q ^-(5, q) \mathcal{Q}^{-}(5,q). Pages: 37–56 Keywords: keywords ovoid; unitary spread; slice Full Text: PDF References 1. Baker, R.D., Ebert, G.L., Wantz, K.L.: Regular hyperbolic fibrations. Adv. Geom. 1(2), 119-144 (2001) 2. Cooperstein, B.N.: Hyperplane sections of Kantor's unitary ovoid. Des. Codes Cryptogr. 23(2), 185- 195 (2001) 3. Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. thesis, Univ. of Meryland, College Park (1974) 4. Dye, R.: Partitions and their stabilizers for line complexes and quadrics. Ann. Mat. Pura Appl. 114(4), 173-194 (1977) 5. Giuzzi, L.: Collineation groups of the intersection of two classical unitals. J. Comb. Des. 9(6), 445- 459 (2001) 6. Hirschfeld, J.W.P.: Finite Projective Spaces of Three Dimension. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1985) 7. Hirschfeld, J.W.P., Thas, J.A.: General Galois Geometries. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991) 8. Kantor, W.M.: Ovoids and translation planes. Can. J. Math. 34(5), 1195-1207 (1982) 9. Kantor, W.M.: Spreads, translation planes and Kerdock sets I. SIAM J. Algebr. Discrete Methods 3(2), 151-165 (1982) 10. Kestenband, B.C.: Unital intersections in finite projective planes. Geom. Dedicata 11(1), 107-117 (1981) 11. Lunardon, G.: Normal spreads. Geom. Dedicata 75(3), 245-261 (1999) 12. Moisio, M.: Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm. Acta Arith. 132(4), 329-350 (2008) 13. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. Pitman Advanced Publishing Program (1984) J Algebr Comb (2011) 33: 37-56 © 1992–2009 Journal of Algebraic Combinatorics © 2012 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition

JOURNAL OF ALGEBRAIC COMBINATORICS

Slices of the unitary spread

Abstract

References

JOURNAL OF
ALGEBRAIC
COMBINATORICS