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

Let d≥3d≥3. Let HH be a d+1d+1-dimensional vector space over GF(2)GF(2) and {e0,…,ed}{e0,…,ed} be a specified basis of HH. We define Supp(t)≔{et1,…,etl}Supp(t)≔{et1,…,etl}, a subset of a specified base for a non-zero vector t=et1+⋯+etlt=et1+⋯+etl of HH, and Supp(0)≔0̸Supp(0)≔0̸. We also define J(t)≔Supp(t)J(t)≔Supp(t) if |Supp(t)||Supp(t)| is odd, and J(t)≔Supp(t)∪{0}J(t)≔Supp(t)∪{0} if |Supp(t)||Supp(t)| is even.For s,t∈Hs,t∈H, let {a(s,t)}{a(s,t)} be elements of H⊕(H∧H)H⊕(H∧H) which satisfy the following conditions: (1) a(s,s)=(0,0)a(s,s)=(0,0), (2) a(s,t)=a(t,s)a(s,t)=a(t,s), (3) a(s,t)≠(0,0)a(s,t)≠(0,0) if s≠ts≠t, (4) a(s,t)=a(s′,t′)a(s,t)=a(s′,t′) if and only if {s,t}={s′,t′}{s,t}={s′,t′}, (5) {a(s,t)|t∈H}{a(s,t)|t∈H} is a vector space over GF(2)GF(2), (6) {a(s,t)|s,t∈H}{a(s,t)|s,t∈H} generate H⊕(H∧H)H⊕(H∧H). Then, it is known that S≔{X(s)|s∈H}S≔{X(s)|s∈H}, where X(s)≔{a(s,t)|t∈H∖{s}}X(s)≔{a(s,t)|t∈H∖{s}}, is a dual hyperoval in PG(d(d+3)/2,2)=(H⊕(H∧H))∖{(0,0)}PG(d(d+3)/2,2)=(H⊕(H∧H))∖{(0,0)}.In this note, we assume that, for s,t∈Hs,t∈H, there exists some xs,txs,t in GF(2)GF(2) such that a(s,t)a(s,t) satisfies the following equation: a(s,t)=∑w∈J(t)a(s,w)+xs,t(a(s,0)+a(s,e0)). Then, we prove that the dual hyperoval constructed by {a(s,t)}{a(s,t)} is isomorphic to either the Huybrechts’ dual hyperoval, or the Buratti and Del Fra’s dual hyperoval.