Abstract
The (0,α)-geometries fully embedded in a projective space are up to a few open cases classified. For (0,α)-geometries fully embedded in an affine space AG(n,q), less is known. The most important model is the so-called linear representation T n-1* (k) of a set k of type {0,1,α +1} with respect to lines in the hyperplane at infinity. We will give a characterization of this model. We also investigate the case where the (0,α)-geometry, fully embedded in AG(n,q), is the dual of a semipartial geometry.
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References
R. C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math., Vol. 13 (1963) pp. 389–419.
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin (1989).
M. Brown, F. De Clerck and M. Delanote, Affine semipartial geometries and projections of quadrics, J. Comb. Theory Ser. A, Vol. 103 (2003) pp. 281–289.
M. R. Brown, Semipartial geometries and generalized quadrangles of order (r, r 2), Bull. Belg. Math. Soc. Simon Stevin, Vol. 5, No. 2–3 (1998) pp. 187–205. Finite geometry and combinatorics, Deinze (1997).
F. Buekenhout and C. Lefevre, Generalized quadrangles in projective spaces, Arch. Math. (Basel), Vol. 25 (1974) pp. 540–552.
I. Debroey and J. A. Thas, On semipartial geometries, J. Combin. Theory Ser. A, Vol. 25 (1978) pp. 242–250.
I. Debroey and J. A. Thas, Semipartial geometries in AG(2; q) and AG(3; q), Simon Stevin, Vol. 51 (1978) pp. 195–209.
F. De Clerck and J. A. Thas, The embedding of (0; α)-geometries in PG(n; q), Part I, Ann. Discrete Math., Vol. 18 (1983) pp. 229–240.
F. De Clerck and H. Van Maldeghem, On linear representations of (α, β)-geometries, European J. Combin., Vol. 15 (1994) pp. 3–11.
F. De Clerck and H. Van Maldeghem, Some classes of rank 2 geometries. In F. Buekenhout (ed.), Handbook of Incidence Geometry, Buildings and Foundations, chapter 10, North-Holland, Amsterdam (1995) pp. 433–475.
N. De Feyter, Classification of (0, 2)-geometries embedded in AG(3, q), Preprint.
N. De Feyter, The embedding of (0; 2)-geometries and semipartial geometries in AG(n, q), Preprint. to appear in Adv. Geom.
M. T. Scafati, k, n-archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri, I, II, Atti Accad. Naz. Lincei Rend., Vol. 40 (1996) pp. 812–818; 1020–1025.
J. A. Thas, Partial geometries in finite affine spaces, Math. Z., Vol. 158, No. 1 (1978) pp. 1–13.
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Clerck, F.D., Delanote, M. On (0,α)-Geometries and Dual Semipartial Geometries Fully Embedded in an Affine Space. Designs, Codes and Cryptography 32, 103–110 (2004). https://doi.org/10.1023/B:DESI.0000029215.28176.62
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DOI: https://doi.org/10.1023/B:DESI.0000029215.28176.62