New quotients of the d-dimensional Veronesean dual hyperoval in PG(2d + 1,2)

Abstract

Let d ≥ 3 . For each e ≥ 1 , Thas and Van Maldeghem constructed a d -dimensional dual hyperoval in PG ( d ( d + 3 ) ∕ 2 , q ) with q = 2 e , called the Veronesean dual hyperoval. A quotient of the Veronesean dual hyperoval with ambient space PG ( 2 d + 1 , q ) , denoted S σ , is constructed by Taniguchi, using a generator σ of the Galois group Gal ( GF ( q d + 1 ) ∕ GF ( q ) ) . In this note, using the above generator σ for q = 2 and a d -dimensional vector subspace H of GF ( 2 d + 1 ) over GF ( 2 ) , we construct a quotient S σ , H of the Veronesean dual hyperoval in PG ( 2 d + 1 , 2 ) in case d is even. Moreover, we prove the following: for generators σ and τ of the Galois group Gal ( GF ( 2 d + 1 ) ∕ GF ( 2 ) ) , S σ above (for q = 2 ) is not isomorphic to S τ , H , S σ , H is isomorphic to S σ , H ′ for any d -dimensional vector subspaces H and H ′ of GF ( 2 d + 1 ) , and S σ , H is isomorphic to S τ , H if and only if σ = τ or σ = τ − 1 . Hence, we construct many new non-isomorphic quotients of the Veronesean dual hyperoval in PG ( 2 d + 1 , 2 ) .