Contents

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 \( 112 \leq m_2(6,3) \leq 136 \).

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. .