On 2-dimensional dual hyperovals of polar type

Abstract

Exploiting an o-polynomial over GF(4), a direct proof is given for the fact that a 2-dimensional dual hyperoval S in PG(5, 4) of unitary polar type is isomorphic to the one exhibited in ([2] p.39). The proof provides enough automorphisms of S to conclude that Aut(S) ≅- M₂₂.2. This gives an account for the exceptional embedding of the Mathieu group M₂₂ into the unitary group PGU₆(2) because the linear part of the automorphism group of a dimensional dual hyperoval of polar type is shown to preserve the associated form modulo the scalar multiplications (cf. Theorem 1).