A Result by Gronchi and Heim

Let m₂(u, b) denote the size of the largest caps in the projective space PG(u, b). In [1, Table 4.3(i)], the following upper bounds on m₂(4, b) are listed:

• m₂(4, 9) ≤ 703, therefore s ≤ 703 for all linear orthogonal arrays OA(9⁵, s,F₉, 3) / [s, s−5, 4]-codes over F₉

• m₂(4, 11) ≤ 1266, therefore s ≤ 1266 for all linear OA(11⁵, s, ℤ₁₁, 3) / [s, s−5, 4]-codes over ℤ₁₁

• m₂(4, 13) ≤ 2107, therefore s ≤ 2107 for all linear OA(13⁵, s, ℤ₁₃, 3) / [s, s−5, 4]-codes over ℤ₁₃

These bounds are obtained by applying the ideas of Gronchi [2] to the results of Heim [3].

References

[1]	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.
[2]	Paolo Gronchi. I blocking sets ed il packing problem. Unione Matematica Italiana. Bollettino. A. Serie VII, 7(2):227–236, 1993.
[3]	U. Heim. On t-blocking sets in projective spaces, 1994. Unpublished manuscript.

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 Gronchi and Heim.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CBoundB91113M5Cap.html