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Journal of Combinatorial Mathematics and Combinatorial Computing

  • Research article

On Complete Caps in the Projective Geometries over \(\mathbb{F}_{3}\) II: New Improvements

J. Barat1, Y. Edel2, R. Hill3, L. Storme4
1JANOS BARAT, Bolyai Institute, University of Szeged, Aradi Vértantk tere 1., 6720, Hungary
2Yves EDEL, University of Heidelberg, Mathematisches Institut der Universitit, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany
3Ray HILL, Department of Computer and Mathematical Sciences, University of Salford, Salford M5 4WT, U.K.
4Dept. of Pure Maths and Computer Algebra, Krijgslaan 281, 9000 Gent, Belgium
  • Published: 31/05/2004

Abstract

Hill, Landjev, Jones, Storme, and Barat proved in a previous article on caps in PG(5, 3) and PG(6,3) that every 53-cap in PG(5, 3) is contained in the 56-cap of Hill and that there exist complete 48-caps in PG(5,3). The first result was used to lower the upper bound on \( m_2(6,3) \) on the size of caps in PG(6, 3) from 164 to 154. Presently, the known upper bound on \( m_2(6, 3) \) is 148. In this article, using computer searches, we prove 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. Computer searches for caps in PG(6,3) which use the computer results of PG(5,3) then lower the upper bound on \( m_2(6,3) \) to \( m_2(6,3) \leq 136 \). So now we know that \( 112 \leq m_2(6,3) \leq 136 \).

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Citation
J. Barat, Y. Edel, R. Hill, L. Storme. On Complete Caps in the Projective Geometries over \(\mathbb{F}_{3}\) II: New Improvements[J], Journal of Combinatorial Mathematics and Combinatorial Computing, Volume 049. 9-31. DOI: .