A Result by Barát, Edel, Hill, and Storme

Let m2(u, b) denote the size of the largest cap in the projective space PG(u, b). In [1], using computer searches, it is proven that every 49-cap in PG(5, 3) is contained in a 56-cap, and that every 48-cap, having a 20-hyperplane with at most 8-solids, is also contained in a 56-cap. These computer results allow a geometrical proof that m2(6, 3) ≤ 147. A computer search for caps in PG(6, 3) which uses the computer results of PG(5, 3) then lowers this bound to m2(6, 3) ≤ 136.

Therefore, for every linear orthogonal array OA(37, s, ℤ3, 3) / linear [s, s−7, 4]-code over 3 we must have s ≤ 136.

References

[1]János Barát, Yves Edel, Raymond Hill, and Leo Storme.
On complete caps in the projective geometries over F3 II: New improvements.
Journal of Combinatorial Mathematics and Combinatorial Computing, 49:9–31, 2004.

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. “A Result by Barát, Edel, Hill, and Storme.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CBoundB3M7Cap.html

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