Minimal Covers of Q +(2 n + 1, q) by ( n - 1)-Dimensional Subspaces
J. Eisfeld
, L. Storme2
and P. Sziklai2
2dagger
DOI: 10.1023/A:1015060407685
Abstract
A t-cover of a quadric Q Q is a set C of t-dimensional subspaces contained in Q Q such that every point of Q Q is contained in at least one element of C.
We consider ( n - 1)-covers of the hyperbolic quadric Q +(2 n + 1, q). We show that such a cover must have at least q n + 1 + 2 q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.
Pages: 231–240
Keywords: covers; partial spreads; quadrics
Full Text: PDF
References
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2. R.C. Bose and R.C. Burton, “A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and McDonald codes,” J. Combin. Theory 1 (1966), 96-104.
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5. J. Eisfeld, L. Storme, T. Szónyi, and P. Sziklai, “Covers and blocking sets of classical generalized quadrangles,” in Proceedings of the Third International Shanghai Conference on Designs, Codes and Finite Geometries (Shanghai, China, May 14-18, 1999). Discrete Math. 238 (2001), 35-51.
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2. R.C. Bose and R.C. Burton, “A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and McDonald codes,” J. Combin. Theory 1 (1966), 96-104.
3. J. Eisfeld, “On smallest covers of finite projective spaces,” Arch. Math. 68 (1997), 77-80.
4. J. Eisfeld, L. Storme, and P. Sziklai, “Minimal covers of the Klein quadric,” J. Combin. Theory, Ser. A 95 (2001), 145-157.
5. J. Eisfeld, L. Storme, T. Szónyi, and P. Sziklai, “Covers and blocking sets of classical generalized quadrangles,” in Proceedings of the Third International Shanghai Conference on Designs, Codes and Finite Geometries (Shanghai, China, May 14-18, 1999). Discrete Math. 238 (2001), 35-51.
6. N. Hamada, “A characterization of some [n, k, d; q]-codes meeting the Griesmer bound using a minihyper in a finite projective geometry,” Discrete Math. 116 (1993), 229-268.
7. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, 2nd edn., Oxford University Press, Oxford, 1998.
8. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991.
9. K. Metsch, “The sets closest to ovoids in Q - (2n + 1, q),” Bull. Belg. Math. Soc. Simon Stevin 5 (1998), 389-392.
10. K. Metsch, “Bose-Burton type theorems for finite projective, affine and polar spaces,” Surveys in Combinatorics, 1999 (Canterbury), London Math. Soc. Lecture Note Ser. 267, Cambridge University Press, Cambridge 1999, pp. 137-166.
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