The eight dimensional ovoids over GF(5)

Charnes, C.; Dempwolff, U.

doi:10.1090/S0025-5718-00-01191-1

The eight dimensional ovoids over GF(5)

Authors: C. Charnes and U. Dempwolff
Journal: Math. Comp. 70 (2001), 853-861
MSC (2000): Primary 51E15, 68R05, 05B25
DOI: https://doi.org/10.1090/S0025-5718-00-01191-1
Published electronically: February 23, 2000
MathSciNet review: 1680855
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

In this article we outline a computer assisted classification of the ovoids in an orthogonal space of the type $\Omega^{+}(8,5)$ .

1. Aart Blokhuis and G. Eric Moorhouse, Some 𝑝-ranks related to orthogonal spaces, J. Algebraic Combin. 4 (1995), no. 4, 295–316. MR 1346886, https://doi.org/10.1023/A:1022477715988
2. Chris Charnes, Quadratic matrices and the translation planes of order 5², Coding theory, design theory, group theory (Burlington, VT, 1990) Wiley-Intersci. Publ., Wiley, New York, 1993, pp. 155–161. MR 1227127
3. Sandro Rajola, An algebraic characterization of sets of type (0,𝑛)₁ in 𝐴𝐺(2,𝑞), J. Geom. 58 (1997), no. 1-2, 158–163. MR 1434189, https://doi.org/10.1007/BF01222937
4. J. H. Conway, P. B. Kleidman, and R. A. Wilson, New families of ovoids in 𝑂⁺₈, Geom. Dedicata 26 (1988), no. 2, 157–170. MR 939461, https://doi.org/10.1007/BF00151667
5. Terry Czerwinski and David Oakden, The translation planes of order twenty-five, J. Combin. Theory Ser. A 59 (1992), no. 2, 193–217. MR 1149895, https://doi.org/10.1016/0097-3165(92)90065-3
6. Terry Czerwinski, The collineation groups of the translation planes of order 25, Geom. Dedicata 39 (1991), no. 2, 125–137. MR 1119648, https://doi.org/10.1007/BF00182289
7. U. Dempwolff, Translation planes of order 27, Des. Codes Cryptogr. 4 (1994), no. 2, 105–121. MR 1268564, https://doi.org/10.1007/BF01578865
8. U. Dempwolff and A. Reifart, The classification of the translation planes of order 16. I, Geom. Dedicata 15 (1983), no. 2, 137–153. MR 737954, https://doi.org/10.1007/BF00147760
9. A. Gunawardena, Classification of ovoids in $\Omega_8^+(4)$ , submitted for publication, 1996.
10. William M. Kantor, Ovoids and translation planes, Canadian J. Math. 34 (1982), no. 5, 1195–1207. MR 675685, https://doi.org/10.4153/CJM-1982-082-0
11. William M. Kantor, Spreads, translation planes and Kerdock sets. I, SIAM J. Algebraic Discrete Methods 3 (1982), no. 2, 151–165. MR 655556, https://doi.org/10.1137/0603015
12. G. Eric Moorhouse, Ovoids from the 𝐸₈ root lattice, Geom. Dedicata 46 (1993), no. 3, 287–297. MR 1220120, https://doi.org/10.1007/BF01263620
13. G. Mason and E. E. Shult, The Klein correspondence and the ubiquity of certain translation planes, Geom. Dedicata 21 (1986), no. 1, 29–50. MR 850561, https://doi.org/10.1007/BF00147528
14. Rudolf Mathon and Gordon F. Royle, The translation planes of order 49, Des. Codes Cryptogr. 5 (1995), no. 1, 57–72. MR 1313377, https://doi.org/10.1007/BF01388504
15. M. Schönert et al., GAP Groups, Algorithms and Programming 3.4.4, Lehrstuhl D für Mathematik, RWTH Aachen, 1997.
16. Ernest Shult, Nonexistence of ovoids in Ω⁺(10,3), J. Combin. Theory Ser. A 51 (1989), no. 2, 250–257. MR 1001266, https://doi.org/10.1016/0097-3165(89)90050-2
17. J. A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1981), no. 1-4, 135–143. MR 608135, https://doi.org/10.1007/BF01447417