Large caps in projective Galois spaces
Jürgen Bierbrauer, Daniel Edel · Ghent University Academic Bibliography (Ghent University) · 2011
If the underlying field is F2, the answer is easy: AG(n,2) is itself a cap of 2n points and it forms up to projective equivalence the unique largest cap in PG(n,2). Assume therefore we work in PG(n,q) or AG(n,q) for q > 2. The canonical models for caps are quadrics of Witt index 1. They yield (q+1)-caps in AG(2,q) (and in PG(2,q)) and obviously these ovals are maximal for odd q. In odd characteristic each oval in PG(2,q) is a conic section (Segre [49,50]). This is not true in characteristic 2, where moreover each oval O is embedded in a unique hyperoval O ∪{N}. Here N is the nucleus, the intersection of all the tangents to O. A hyperoval is a (q+2)-cap and this is maximal. The hyperovals are described by a special kind of permutation polynomials. This is an active line of research, see the survey [38]. In PG(3,q) an elliptic quadric is a (q2 + 1)-cap. This is maximal for all q > 2 (see Bose [13] and Qvist [47]). Its affine part is a q2-cap in AG(3,q) and this is maximal. In characteristic 2 the Tits ovoids form another family of (q2 + 1)-caps in PG(3,q), see [55]. They may be considered classical as they admit a family of classical groups, the Suzuki groups B2(q) for q = 22m+1, as groups of automorphisms. ∗Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA. E-mail address: [email protected] †Ghent University, Department of Mathematics, Krijgslaan 281 S22, B–9000 Gent, Belgium. E-mail address: [email protected]. The research of this author takes place within the project Linear codes and cryptography of the Research Foundation – Flanders (FWO) (Project nr. G.0317.06), and is supported by the Interuniversitary Attraction Poles Programme Belgian State Belgian Science Policy: project P6/26Bcrypt.