Vol. 4, No. 1, 2006

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Caps in projective Hjelmslev spaces over finite chain rings of nilpotency index $2$

Thomas Honold and Ivan N. Landjev

Vol. 4 (2006), No. 1, 13–25

We investigate caps in the projective Hjelmslev geometries PHG(RRk) over chain rings R with |R| = q2, RradRFq. We present a geometric construction for caps using ovoids in the factor geometry PG(3,q) as well as an algebraic construction that makes use of the Teichmüller group of units in the Galois extension of certain chain rings. We prove upper bounds on the size of a maximal cap in PHG(RR4). It has an order of magnitude q4. This bound extends to higher dimensions, but gives the rather rough estimate q2k4.

Mathematical Subject Classification 2000
Primary: 51E21, 51E22, 51E26, 94B05
Received: 8 April 2006
Accepted: 23 October 2006
Thomas Honold
Ivan N. Landjev