Blocking sets in PG (2, q n ) from cones of PG (2 n , q )
Francesco Mazzocca
and Olga Polverino
Seconda Universit`a degli Studi di Napoli Dipartimento di Matematica via Vivaldi 43, I-81100 Caserta, Italy
DOI: 10.1007/s10801-006-9102-y
Abstract
Let Ω and [ `( B)] {\bar B} be a subset of Σ = PG(2 n - 1, q) and a subset of PG(2 n, q) respectively, with Σ \subset PG(2 n, q) and [ `( B)] Ë S {{\bar B}\not\subset Σ} . Denote by K the cone of vertex Ω and base [ `( B)] {\bar B} and consider the point set B defined by
| B=( K\ S) È{ X Ĩ {\S} : X Ç K \textonesuperior Æ}, B=\big(K{\setminus}Σ\big) \cup \{X\in \S\, : \, X\cap K\neq \emptyset\},
Pages: 61–81 Keywords: keywords blocking set; André/bruck-Bose representation; ovoid Full Text: PDF References1. J. André: \ddot Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z., 60 (1954), 156-186.
2. S. Ball: On ovoids of O(5, q), Adv. Geom., 4 n.1 (2004), 1-7. 3. S. Ball, P. Govaerts and L. Storme: On ovoids of parabolic quadrics, Des. Codes Cryptogr., to appear. 4. S. G. Barwick, L. R. A. Casse and C. T. Quinn: The André/Bruck and Bose representation in P G(2h, q) : unitals and Baer subplanes, Bull. Belg. Math. Soc. Simon Stevin, 7 (2000) no.2, 173- 197. 5. L. Berardi and F. Eugeni: Blocking sets e teoria dei giochi: origini e problematiche, Atti Sem. Mat. Fis. Univ. Modena, 34 (1988), 165-196. 6. A. Beutelspacher: Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata, 9 (1980), 425-449. 7. A. Blokhuis: On the size of a blocking set in P G(2, p), Combinatorica, 14 (1994), 273- 276. 8. A. Blokhuis: Blocking sets in Desarguesian planes, in Paul Erd\ddot os is Eighty, Volume 2, Bolyai Soc. Math. Studies 2, Bolyai Society, Budapest, (1996), 133-155. 9. R. H. Bruck and R. C. Bose: The construction of translation planes from projective spaces, J. Algebra, 1 (1964), 85-102. 10. R. H. Bruck and R. C. Bose: Linear representations of projective planes in projective spaces, J. Algebra, 4 (1966), 117-172. 11. A. A. Bruen: Baer subplanes and blocking sets, Bull. Amer. Math. Soc., 76 (1970), 342- 344. 12. A. A. Bruen and J. A. Thas: Hyperplane Coverings and Blocking Sets, Math. Z., 181 (1982), 407- 409. 13. F. Buekenhout: Existence of unitals in finite translation planes of order q2 with a kernel of order q, Geom. Dedicata, 5 (1976), 189-194. 14. A. Cossidente, A. Gács, C. Mengyán, A. Siciliano, T. Sz\Acute\Acute onyi and Zs. Weiner: On large minimal blocking sets in P G(2, q), J. Combin. Des., 13 n.1 82005, 25-41. 15. A. Gunawardena and G. E. Moorhouse: The non-existence of ovoids in O9(q), European J. Combin., 18 (1997), 171-173. 16. U. Heim: Proper blocking sets in projective spaces, Discrete Math., 174 (1997), 167-176. 17. J. W. P. Hirschfeld: Projective geometries over finite fields, Clarendon Press, Oxford, 1979, 2nd Edition, 1998. 18. T. Ill`es, T. Sz\Acute\Acute onyi and F. Wettl: Blocking sets and maximal strong representative systems in finite projective planes, Mitt. Math. Sem. Giessen, 201 (1991), 97-107. 19. S. Innamorati and A. Maturo: On irreducible blocking sets in projective planes, Ratio Math., 208/209 (1991), 151-155. 20. W. M. Kantor: Ovoids and translation planes, Canad. J. Math., 34 (1982), 1195-1207. 21. G. Lunardon: Normal spreads, Geom. Dedicata, 75 (1999), 245-261. 22. G. Lunardon: Linear k - blocking sets, Combinatorica, 21(4) (2001), 571-581. 23. G. Lunardon and O. Polverino: Linear blocking sets: a survey, Finite fields and applications, Springer, 2001, 356-362. 24. O. Polverino: On normal spreads, Preprint (2003). 25. B. Segre: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76. 26. L. Storme and Zs.Weiner: Minimal blocking sets in P G(n, q), n \geq 3., Designs, Codes and Cryptography, 21 (2000), 235-251. 27. P. Sziklai and T. Sz\Acute\Acute onyi: Blocking sets and algebraic curves, Rend. Circ. Mat. Palermo, 51 (1998), 71-86. 28. T. Sz\Acute\Acute onyi: Note on the existence of large minimal blocking sets in Galois planes, Combinatorica, 12 (1992), 227-235. Springer 29. T. Sz\Acute\Acute onyi: Blocking sets in finite planes and spaces, Ratio Math., 5 (1992), 93-106. 30. T. Sz\Acute\Acute onyi, A. Gács and Zs. Weiner: On the spectrum of minimal blocking sets in P G(2, q), J. Geom., 76 n.112 (2003), 256-281. 31. G. Tallini: Blocking sets with respect to planes in P G(3, q) and maximal spreads of an non singular quadric in P G(4, q), Mitt. Math. Sem. Giessen, 201 (1991), 97-107. 32. J. A. Thas: Ovoids and spreads of finite classical polar spaces, Geom. Dedicata, 174 (1981), 135-143. 33. J. Tits: Ovoides et groupes de Suzuki, Arch. Math., 13 (1962), 187-198.
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