On Hadamard designs associated with a hyperoval
A hyperoval in a projective plane \(\pi\) of order \(q\) is a subset \(\Omega\) of \(q + 2\) points in \(\pi\), no three of which are collinear. Every line of \(\pi\) necessarily meets a hyperoval in zero or two points, the former lines being called exterior lines and the latter ones secant lines. If \(P\) is any point of \(\pi \backslash \Omega\), let \(S(P) = \{Q \in \pi \backslash \Omega : Q \neq P\) and the line \(PQ\) is secant to \(\Omega\} \cup \{P\}\). Suppose \(q = 2^d\), where \(d \geq 3\). Let \(N\) be a point of \(\Omega\), and let \(t\) be a line through \(N\). For every other line \(s\) through \(N\), let \(D_{t,s}\) denote the incident structure whose points are those of \(t \backslash (\Omega \cap t)\) and whose blocks are sets \(S(P) \cap t\) as \(P\) varies over \(s \backslash (s \cap \Omega)\). One can show that \(D_{t,s}\) has \(q-1\) points and blocks, and each block is incident with \({q \over 2} - 1\) points, although \(D_{t,s}\) is not necessarily a 2- design. In the paper under review it is shown that if \(\Omega\) is a translation hyperoval with respect to the line \(t\), then \(D_{t,s}\) is isomorphic to the 2-design of points and hyperplanes of \(PG (d - 1,2)\) for every line \(s\) through \(N,s \neq t\). In particular, \(D_{t,s}\) will be a Hadamard design with parameters \(v = q - 1\), \(k = {q \over 2} - 1\), and \(\lambda = {q \over 4} - 1\).
scientific article; zbMATH DE number 785152