## A Result by Gronchi and Heim

Let *m*_{2}(*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 *m*_{2}(4, *b*) are listed:

*m*_{2}(4, 9) ≤ 703, therefore*s*≤ 703 for all linear orthogonal arrays OA(9^{5},*s*,**F**_{9}, 3) / [*s*,*s*−5, 4]-codes over**F**_{9}*m*_{2}(4, 11) ≤ 1266, therefore*s*≤ 1266 for all linear OA(11^{5},*s*, ℤ_{11}, 3) / [*s*,*s*−5, 4]-codes over ℤ_{11}*m*_{2}(4, 13) ≤ 2107, therefore*s*≤ 2107 for all linear OA(13^{5},*s*, ℤ_{13}, 3) / [*s*,*s*−5, 4]-codes over ℤ_{13}

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Unpublished manuscript.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.”
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