Abstract
The incidence structure NQ+(3, q) has points the points not on a non-degenerate hyperbolic quadric Q+(3, q) in PG(3, q), and its lines are the lines of PG(3, q) not containing a point of Q+(3, q). It is easy to show that NQ+(3, q) is a partial linear space of order (q, q(q−1)/2). If q is odd, then moreover NQ+(3, q) satisfies the property that for each non-incident point line pair (x,L), there are either (q−1)/2 or (q+1)/2 points incident with L that are collinear with x. A partial linear space of order (s, t) satisfying this property is called a ((q−1)/2,(q+1)/2)-geometry. In this paper, we will prove the following characterization of NQ+(3,q). Let S be a ((q−1)/2,(q+1)/2)-geometry fully embedded in PG(n, q), for q odd and q>3. Then S = NQ+(3, q).
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References
S. Ball A. Blokhuis F. Mazzocca (1997) ArticleTitleMaximal arcs in Desarguesian planes of odd order do not exist Combinatorica. 17 IssueID1 31–41 Occurrence Handle10.1007/BF01196129 Occurrence Handle98h:51014
A. Barlotti (1956) ArticleTitleSui {k;n}-archi di un piano lineare finito Boll. Un. Mat. Ital. 11 553–556 Occurrence Handle0072.38103 Occurrence Handle18,666e
S. Cauchie, Full embeddings of (α, β)-geometries in projective spaces. Part II, Discrete Math. Vol. 266, No. 1–3, (2003) pp. 53–183. In Proceedings of the 18th British Combinatorial Conference.
S. Cauchie F. Clerck ParticleDe N. Hamilton (2002) ArticleTitleFull embeddings of (α, β)-geometries in projective spaces Europ. J. Comb. 23 IssueID6 635–646
F. De Clerck and J. A. Thas, The embedding of (0, α)-geometries in PG(n, q). I. Combinatorics ’81 (Rome, 1981) pp. 229–240. North-Holland. Amsterdam (1983).
R. Di Gennaro, N. Durante and D. Olanda, A characterization of the family of lines external to a hyperbolicquadric of (3, q), Submitted to J. Geom.
C. Lefèvre-Percsy (1980) ArticleTitleAn extension of a theorem of G. Tallini J. Combin. Theory Ser. A. 29 IssueID3 297–305 Occurrence Handle0457.51019 Occurrence Handle82c:05030
G. Tallini (1956) ArticleTitleSulle k-calotte di uno spazio lineare finito Ann. Mat. Pura Appl. 42 IssueID4 119–164 Occurrence Handle0074.15304 Occurrence Handle19,55f
G. Tallini (1957) ArticleTitleCaratterizzazione grafica delle quadriche ellittiche negli spazi finiti Rend. Mat. e Appl. 16 IssueID5 328–351 Occurrence Handle20 #865
J. A. Thas I. Debroey F. Clerck ParticleDe (1984) ArticleTitleThe embeddings of (0, α)-geometries in PG(n, q): Part II Discrete Math. 51 283–292 Occurrence Handle10.1016/0012-365X(84)90009-8 Occurrence Handle86g:51010
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J. W. P. Hirschfeld
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Cauchie, S. A Characterization of the Complement of a Hyperbolic Quadric in PG (3, q), for q Odd. Des Codes Crypt 38, 195–208 (2006). https://doi.org/10.1007/s10623-005-6341-1
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DOI: https://doi.org/10.1007/s10623-005-6341-1