Bilinear dual hyperovals from binary commutative presemifields

Abstract

For a binary commutative presemifield S with an element c∈S, we can construct a bilinear dual hyperoval Sc(S) if c satisfies some conditions. Let c1∈S1 and c2∈S2 for commutative presemifields S1 and S2, and assume c1≠1 or c2≠1. Then the dual hyperovals Sc1(S1) and Sc2(S2) are isomorphic if and only if S1 and S2 are isotopic with some relation between c1 and c2 induced by the isotopy. For the Kantor commutative presemifield S=(F,+,∘) with c∈Fn⊂F, the dual hyperoval Sc(S) exists if and only if Tr(c)=1, where Tr is the absolute trace of Fn. The dual hyperovals Sc1(S1) and Sc2(S2) for the Kantor commutative presemifields S1 and S2 are (under some conditions) isomorphic if and only if S1 and S2 are isotopic with c1σ=c2, where σ is the field automorphism of F defined by the isotopy.