Bound for Caps in PG(4, b)

Let m2(u, b) denote the size of the largest caps in the projective space PG(u, b). Then, for all b ≥ 7, we have

m2(4, b) ≤ b3 – b2 + \begin{displaymath}\begin{cases}8b−13 & \textrm{if $b$\ is odd}\\ 6b−3 & \textrm{if $b$\ is even.}\end{cases}\end{displaymath}

The result for odd b is due to [1], the result for even b to [2] and [3]. See also [4, Table 4.3(i)].

References

[1]Adriano Barlotti.
Some topics in finite geometrical structures.
Mimeo Series 439, Institute of Statistics, Univ. of North Carolina, 1965.
[2]Jin-Ming Chao.
On the size of a cap in PG(n, q) with q even and n ≥ 3.
Geometriae Dedicata, 74(1):91–94, January 1999.
doi:10.1023/A:1005095418152
[3]James W. P. Hirschfeld and Joseph A. Thas.
Linear independence in finite spaces.
Geometriae Dedicata, 23(1):15–31, May 1987.
doi:10.1007/BF00147388
[4]James W. P. Hirschfeld and Leo Storme.
The packing problem in statistics, coding theory and finite projective spaces: Update 2001.
In Finite Geometries, volume 3 of Developments in Mathematics, pages 201–246. Kluwer Academic Publishers, 2001.

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Bound for Caps in PG(4, b).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CBoundCapM5.html

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