Switching sets in 𝑃𝐺(3,𝑞)

Bruen, A.; Silverman, R.

doi:10.1090/S0002-9939-1974-0333946-8

Switching sets in ${\rm PG}(3,\,q)$

Authors: A. Bruen and R. Silverman
Journal: Proc. Amer. Math. Soc. 43 (1974), 176-180
MSC: Primary 50D30; Secondary 05BXX
DOI: https://doi.org/10.1090/S0002-9939-1974-0333946-8
MathSciNet review: 0333946
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Abstract: In this note, we are mainly concerned with partial spreads of which cover the same points and have no line in common. Setting $\vert U\vert = \vert V\vert = t$ , we show that if then $t \geqq \max (q + 2,2q - 2)$ . Certain applications of this result to (0, 1) matrices and to translation planes are then discussed.

[1] R. H. Bruck, Finite nets. II. Uniqueness and imbedding, Pacific J. Math. 13 (1963), 421–457. MR 154824
[2] A. Bruen, Unimbeddable nets of small deficiency, Pacific J. Math. 43 (1972), 51–54. MR 380614
[3] P. Dembowski, Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Springer-Verlag, Berlin-New York, 1968. MR 0233275
[4] David A. Foulser, Replaceable translation nets, Proc. London Math. Soc. (3) 22 (1971), 235–264. MR 0291935, https://doi.org/10.1112/plms/s3-22.2.235
[5] J. W. P. Hirschfeld, Classical configurations over finite fields. I. The double- six and the cubic surface with 27 lines, Rend. Mat. e Appl. (5) 26 (1967), 115–152. MR 233272
[6] H. J. Ryser, Combinatorial configurations, SIAM J. Appl. Math. 17 (1969), 593–602. MR 256904, https://doi.org/10.1137/0117056