Reed-Muller Codes And Hadamard Designs From Ovals

From any projective plane \(\Pi\) of even order \(n\) with an oval (\((n+2)\)-arc), a Hadamard \(3\)-design on \(n^2\) points can be defined using a well-known construction. If \(\Pi\) is Desarguesian with \(n = 2^m\) and the oval is regular (a conic plus nucleus) then
it is shown that the binary code of the Hadamard \(3\)-design contains a copy of the first-order Reed-Muller code of length \(2^{2m}\).