Geodesic
Ovoids and Translation Planes
Canadian journal of mathematics, Tome 34 (1982) no. 5, pp. 1195-1207

Voir la notice de l'article provenant de la source Cambridge University Press

An ovoid in an orthogonal vector space V of type Ω+(2n, q) or Ω(2n – 1, q) is a set Ω of q n–1 + 1 pairwise non-perpendicular singular points. Ovoids probably do not exist when n > 4 (cf. [12], [6]) and seem to be rare when n = 4. On the other hand, when n = 3 they correspond to affine translation planes of order q 2, via the Klein correspondence between PG(3, q) and the Ω+(6, q) quadric.In this paper we will describe examples having n = 3 or 4. Those with n = 4 arise from PG(2, q 3), AG(2, q 3), or the Ree groups. Since each example with n = 4 produces at least one with n = 3, we are led to new translation planes of order q 2. 
Kantor, William M. Ovoids and Translation Planes. Canadian journal of mathematics, Tome 34 (1982) no. 5, pp. 1195-1207. doi: 10.4153/CJM-1982-082-0
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[1] 1. Carlitz, L., A theorem on permutations in a finite field, Proc. AMS 11 (1960), 456–459./ Google Scholar

[2] 2. Dembowski, P., Finite geometries (Springer, Berlin-Heidelberg-New York, 1968./ Google Scholar

[3] 3. Dye, R. H., Partitions and their stabilizers for line complexes and quadrics, Annali di Mat. 114 (1977), 173–194./ Google Scholar

[4] 4. Kantor, W. M., Spreads, translation planes and Kerdock sets I, SI A M J. Alg. Disc. Meth. 3 (1982), 151–165./ Google Scholar

[5] 5. Kantor, W. M., Spreads, translation planes and Kerdock sets II, to appear in Siam J. Alg. Disc. Meth. Google Scholar

[6] 6. Kantor, W. M., Strongly regular graphs defined by spreads, Israel J. Math. 41 (1982), 298–312./ Google Scholar

[7] 7. Kantor, W. M. and Liebler, R. A., The rank 3 permutation representations of the finite classical groups, Trans. AMS 271 (1982), 1–71./ Google Scholar

[8] 8. Knuth, D. E., Finite semifields and projective planes, J. Algebra 2 (1965), 182–217./ Google Scholar

[9] 9. H., Lùneburg, Translation planes (Springer, New York, 1980./ Google Scholar

[10] 10. Ostrom, T. G., The dual Lùneburg planes, Math. Z. 97 (1966), 201–209./ Google Scholar

[11] 11. Patterson, N. J., A four-dimensional Kerdock set over GF(S), J. Comb. Theory (A) 20 (1976), 365–366./ Google Scholar

[12] 12. Thas, J. A., Ovoids and spreads of finite classical polar spaces (to appear in Geom. Ded.). Google Scholar

[13] 13. Tits, J., Sur la trialité et certains groupes qui s'en déduisent, Publ. Math. I.H.E.S. 2 (1959), 14–60. Google Scholar

[14] 14. Tits, J., Les groupes simples de Suzuki et de Ree, Sém. Bourbaki 210 (1960/61)./ Google Scholar

[15] 15. Walker, M., A class of translation planes, Geom. Ded. 5 (1976), 135–146./ Google Scholar

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