On the dual of the dual hyperoval from APN function f(x)=x3+Tr(x9)

Abstract

AbstractUsing a quadratic APN function f on GF(2d+1), Yoshiara (2009) [15] constructed a d-dimensional dual hyperoval Sf in PG(2d+1,2). In Taniguchi and Yoshiara (2005) [13], we prove that the dual of Sf, which we denote by Sf⊥, is also a d-dimensional dual hyperoval if and only if d is even. In this note, for a quadratic APN function f(x)=x3+Tr(x9) on GF(2d+1) by Budaghyan, Carlet and Leander (2009) [2], we show that the dual Sf⊥ and the transpose of the dual Sf⊥T are not isomorphic to the known bilinear dual hyperovals if d is even and d⩾6