Oval Designs in Desarguesian Projective Planes

Authors:
Laurel L. Carpenter
Affiliations:
Clemson University, Clemson SC 29634/ llcarpe@prism.clemson.edu
Venue:
Designs, Codes and Cryptography - Special issue containing papers presented at the Second Upper Michigan Combinatorics Workshop on Designs, Codes and Geometries
Year:
1996

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Designs, Codes and Cryptography

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Abstract

Given any projective plane \Pi of even order qcontaining a hyperoval \cOv, a Steiner 2- ({{q}\choose{2}},\frac{q}{2},1)design can be constructed. The 2-rank of this design is boundedabove by { rank}_2(\Pi)-q-1. Using a result ofBlokhuis and Moorhouse bm94, we show that this bound is met when\Pi is desarguesian and \cOv is regular.We also show that the block graph of the Steiner 2-design inthis case produces a Hadamard design which is such that the binarycode of the associated 3-design contains a copy of the first-orderReed-Muller code of length 2^{2m}, where q=2^m.