On some d-dimensional dual hyperovals in PG(d(d+3)/2, 2)

Let d >= 3. Let H be a d + 1-dimensional vector space over GF(2) and {e(0),..., e(d)} be a specified basis of H. We define Supp(t) := {e(t1),....... e(t1)}, a subset of a specified base for a non-zero vector t = e(t1) + ... +e(t1) of H, and Supp(0) := empty set. We also define J (t) := Supp(t) if vertical bar Supp(t)vertical bar is odd, and J(t) := Supp(t) boolean OR {0} if vertical bar Supp(t)vertical bar is even. For s, t is an element of H, let {a(s, t)} be elements of H circle plus (H boolean AND H) which satisfy the following conditions: (1) a(s, s) = (0, 0), (2) a(s, t) = a(t, s), (3) a(s, t) not equal (0, 0) ifs s not equal t, (4) a(s, t) = a(s', t') if and only if {s, t} = {s', t}, (5) {a(s, t)vertical bar t is an element of H} is a vector space over GF(2), (6) {a(s, t)vertical bar s. t is an element of H} generate H circle plus (H boolean AND H). Then, it is known that S := {X(s)vertical bar s is an element of H}, where X(s) := {a(s, t)vertical bar t is an element of H \{s}}, is a dual hyperoval in PG(d(d + 3)/2, 2) = (H (circle plus (H boolean AND H))\{(0, 0)}. In this note, we assume that. for s. t is an element of H, there exists some x(s, t) in GF(2) such that a(s, t) satisfies the following equation: a(s, t) = Sigma(w is an element of J(t)) a(s, w)) + x(s, t) (a(s, 0) + a(s, e(0))). Then, we prove that the dual hyperoval constructed by {a(s, t)} is isomorphic to either the Huybrechts' dual hyperoval, or the Buratti and Del Fra's dual hyperoval. (C) 2009 Elsevier Ltd. All rights reserved.