Vol. 4, No. 1, 2006

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Large caps with free pairs in dimensions five and six

Jeffrey Bryan Farr and Petr Lisonek

Vol. 4 (2006), No. 1, 69–88

A cap in PG(N,q) is said to have a free pair of points if any plane containing that pair contains at most one other point from the cap. In an earlier paper we determined the largest size of caps with free pairs for N = 3 and 4. In this paper we use product constructions to prove similar results in dimensions 5 and 6 that are asymptotically as large as possible. If q > 2 is even, we determine exactly the largest size of a cap in PG(5,q) with a free pair. In PG(5,3) we give constructions of a maximal size 42-cap having a free pair and of the complete 48-cap that contains it. Additionally, we give some sporadic examples in higher dimensions.

cap, free pair, Galois space
Mathematical Subject Classification 2000
Primary: 51E22
Received: 8 June 2006
Accepted: 20 October 2006
Jeffrey Bryan Farr
Petr Lisonek