On the dual of the dual hyperoval from APN function [formula omitted

Abstract

Using a quadratic APN function f on GF ( 2 d + 1 ) , Yoshiara (2009) [15] constructed a d-dimensional dual hyperoval S f in PG ( 2 d + 1 , 2 ) . In Taniguchi and Yoshiara (2005) [13], we prove that the dual of S f , which we denote by S f ⊥ , is also a d-dimensional dual hyperoval if and only if d is even. In this note, for a quadratic APN function f ( x ) = x 3 + Tr ( x 9 ) on GF ( 2 d + 1 ) by Budaghyan, Carlet and Leander (2009) [2], we show that the dual S f ⊥ and the transpose of the dual S f ⊥ T are not isomorphic to the known bilinear dual hyperovals if d is even and d ⩾ 6 .