Some families of complete caps of quadrics and symplectic polarities over GF(8)

✍ Scribed by R. H. Dye

• Publisher: Springer
• Year: 1996
• Tongue: English
• Weight: 673 KB
• Volume: 60
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00147366

✦ Synopsis

A cap on a quadric is a set of its points whose pairwise joins are all chords. A cap is complete if it is not part of a larger one. The only field for which all complete quadric caps are known is GF(2). Those caps are small; the biggest for each quadric is of order the dimension of the ambient space. Apart from information about ovoids in dimensions at most 7, little else is known. Here, the evidence is increased by providing caps over GF(2~), c~ odd, which, if c~ > 1, have size of order the dimension cubed. In particular, complete caps are obtained for the quadrics Q2m(8), + Qsk+7(8), Q~+3(8), Q+k+1(8) and Q~+5(8). These caps on Q+k+7(8) and Q~+3(8) are complete on any Qn (8) of which their quadrics are sections; so is that of Q4l+2 (8) for any Q2,~ (8) of which Q4~+2( 8) is a section with the same kernel. From the correspondence with Q2~ (8) complete caps are obtained for symplectic polarities over GF(8).