On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Abstract

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval ${\mathcal{T}}_\sigma(V)$ over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals ${\mathcal{T}}_\sigma(V)$ and ${\mathcal{T}}_\tau(V)$ in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ  = τ or (2) σ τ  = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.