Oval Designs in Desarguesian Projective Planes

Given any protective plane of even order q containing a hyperoval , a Steiner design can be constructed. The 2-rank of this design is bounded above by rank₂() – q – 1. Using a result of Blokhuis and Moorhouse 3], we show that this bound is met when is desarguesian and is regular. We also show that the block graph of the Steiner 2-design in this case produces a Hadamard design which is such that the binary code of the associated 3-design contains a copy of the first-order Reed-Muller code of length 2^2m , where q = 2 ^m .