The radical of binary dimensional dual hyperovals

Abstract

Let S be a dimensional dual hyperoval of rank n over F2. We introduce and study the radical P(S), which is a subspace of the ambient space U(S) of S invariant under the automorphism group of S. For the vast majority of the known dimensional dual hyperovals we have P(S)=U(S). Interesting is the case of proper radicals, i.e. P(S)≠U(S). Starting point of our investigations is a result of the second author [10, Thm. 1], [7, Thm. 3.6] (Theorem 1.2 below) which characterizes alternating dual hyperovals by the property that S splits over P(S). This Theorem is extended by Theorem 1.3 where we characterize dimensional dual hyperovals S with dim⁡U(S)−dim⁡P(S)=rank(S)−1. Moreover we will show (Theorem 4.6) that a proper radical implies that this dimensional dual hyperoval is a disjoint union of subDHOs of smaller rank. The notion of “disjoint union of subDHOs” has been introduced by Yoshiara [17]. Some theory on dimensional dual hyperovals with proper radicals is developed. Our paper also provides some computational results on dual hyperovals of small rank with a proper radical. These calculations indicate — though dual hyperovals with a proper radical seem to be scarce — that the number of these hyperovals is steadily growing as function of the rank.