Cyclic caps in \(\mathrm{PG}(3,q)\). (English) Zbl 0834.51005
A \(k\)-cap \(K\) on \(\mathrm{PG}(n,q)\) is a set of \(k\) points, no three of which are collinear. \(K\) is complete if it cannot be extended to a \(k + 1\) cap. If \(K\) is invariant under a cyclic subgroup (which acts regularly on \(K)\) of \(\mathrm{PGL}(n + 1,q)\), then \(K\) is cyclic.
The authors show that if \(q\) is even, the only complete cyclic \(k\)-caps in \(\mathrm{PG}(3,q)\) are elliptic quadrics. If \(q\) is odd then, beside the elliptic quadrics, there are cyclic \(k\)-caps containing \(\tfrac12 k\) points of two disjoint elliptic quadrics or two disjoint hyperbolic quadric. There exists cyclic \(k\)-caps stabilized by a transitive cyclic group which fixes exactly one point and one plane.
The cyclic group is represented by a 4 by 4 matrix. The results come from an analysis of the eigenvalues of the matrix and the fixed points or planes of the resulting transformation.
The authors show that if \(q\) is even, the only complete cyclic \(k\)-caps in \(\mathrm{PG}(3,q)\) are elliptic quadrics. If \(q\) is odd then, beside the elliptic quadrics, there are cyclic \(k\)-caps containing \(\tfrac12 k\) points of two disjoint elliptic quadrics or two disjoint hyperbolic quadric. There exists cyclic \(k\)-caps stabilized by a transitive cyclic group which fixes exactly one point and one plane.
The cyclic group is represented by a 4 by 4 matrix. The results come from an analysis of the eigenvalues of the matrix and the fixed points or planes of the resulting transformation.
Reviewer: T. G. Ostrom (Pullman)
MSC:
| 51E22 | Linear codes and caps in Galois spaces |
Software:
AXIOMReferences:
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