Oval designs in Desarguesian projective planes
From a given hyperoval \({\mathcal O}\) in a projective plane \(\Pi\) of even order \(q\) two designs can easily be obtained. The first is the oval design \({\mathcal W} (\Pi, {\mathcal O})\) which is a resolvable Steiner \(2- ({q \choose 2}, {q \over 2},1)\) design whose points are the exterior lines of \({\mathcal O}\) in \(\Pi\) and whose blocks are the points of \(\Pi\) which are not in \({\mathcal O}\), with incidence inherited from the projective plane. (To set the historical record straight, this construction is correctly attributed to \textit{R. C. Bose} and \textit{M. S. Shrikhande}, but the citation should be to a much earlier paper [Trans. Am. Math. Soc. 95, 191-209 (1960; Zbl 0093.31904)]. In fact, Bose claims to have known of the result since 1949, as reported in a paper by \textit{E. Seiden} [Sankhya, Ser. A 25, 393-394 (1963; Zbl 0124.10803)], who independently rediscovered the construction.) The second design is a Hadamard 3-design, denoted by \({\mathcal H} (\Pi, {\mathcal O})\), with parameters \(3- (q^2, {q^2 \over 2}, {q^2 \over 4} - 1)\) obtained as the unique extension of the Hadamard 2-design whose point set \({\mathcal P}\) is the set of points of \(\Pi\) not on \({\mathcal O}\) and whose blocks are, for each point \(P\in {\mathcal P}\), \(B_P = \{Q\in {\mathcal P} \mid QP\) is a secant of \({\mathcal O}\) in \(\Pi\} \cup \{P\}\). The binary codes of these designs are obtained by taking the block-point incidence matrix of the design as a generator matrix over the binary field, \(F_2\). The dimension of such a code is called its 2-rank. It is known that if the order of the projective plane is \(q= 2^m\), then the 2-rank of the code of the oval design for any hyperoval in that plane is bounded from above by \(3^m-2^m\). The author shows that if the plane is a Desarguesian plane, i.e., \(PG(2,q)\) and the hyperoval is a regular hyperoval (a conic together with its nucleus), then this bound is obtained. It is conjectured that this is true for any hyperoval in a Desarguesian plane (and verified by computer for all known hyperovals in Desarguesian planes of order \(q\leq 64\).) Another interesting question asked about the codes of these designs is whether or not the code of the Hadamard 3-design contains a copy of the first-order Reed-Muller code \({\mathcal R} (1,2m)\) if the plane has order \(q= 2^m\). The author proves the affirmative answer again in the case that the plane is Desarguesian and the hyperoval is regular. The paper includes a table of 2-ranks for the codes of these designs (and some related codes) for Desarguesian planes of order \(q\), \(4\leq q \leq 32\), and all the hyperovals in these planes.
scientific article; zbMATH DE number 934917