On some d-dimensional dual hyperovals in [formula omitted

Abstract

Let d ≥ 3 . Let H be a d + 1 -dimensional vector space over G F ( 2 ) and { e 0 , … , e d } be a specified basis of H . We define S u p p ( t ) ≔ { e t 1 , … , e t l } , a subset of a specified base for a non-zero vector t = e t 1 + ⋯ + e t l of H , and S u p p ( 0 ) ≔ 0̸ . We also define J ( t ) ≔ S u p p ( t ) if | S u p p ( t ) | is odd, and J ( t ) ≔ S u p p ( t ) ∪ { 0 } if | S u p p ( t ) | is even. For s , t ∈ H , let { a ( s , t ) } be elements of H ⊕ ( H ∧ H ) which satisfy the following conditions: (1) a ( s , s ) = ( 0 , 0 ) , (2) a ( s , t ) = a ( t , s ) , (3) a ( s , t ) ≠ ( 0 , 0 ) if s ≠ t , (4) a ( s , t ) = a ( s ′ , t ′ ) if and only if { s , t } = { s ′ , t ′ } , (5) { a ( s , t ) | t ∈ H } is a vector space over G F ( 2 ) , (6) { a ( s , t ) | s , t ∈ H } generate H ⊕ ( H ∧ H ) . Then, it is known that S ≔ { X ( s ) | s ∈ H } , where X ( s ) ≔ { a ( s , t ) | t ∈ H ∖ { s } } , is a dual hyperoval in P G ( d ( d + 3 ) / 2 , 2 ) = ( H ⊕ ( H ∧ H ) ) ∖ { ( 0 , 0 ) } . In this note, we assume that, for s , t ∈ H , there exists some x s , t in G F ( 2 ) such that a ( s , t ) satisfies the following equation: a ( s , t ) = ∑ w ∈ 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.